j. differential geometry

58 (2001) 1-86

MINIMIZING AREA AMONG LAGRANGIAN SURFACES: THE MAPPING PROBLEM

R. SCHOEN & J. WOLFSON

Abstract

This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the lagrangian surfaces of a Kähler surface, or, more generally, a symplectic 4-manifold with suitable metric, and study its critical points and in particular its minimizers. We apply this study to the problem of finding canonical representatives of the lagrangian homology (that part of the homology generated by lagrangian cycles). We show that the lagrangian homology of a Kähler surface (or of a symplectic 4-manifold) is generated by minimizing lagrangian surfaces that are branched immersions except at finitely many singular points. We precisely describe the structure of these singular points. In particular, these singular points are represented by lagrangian cones with an associated local Maslov index. Only those cones of Maslov index 1 or −1 may be area minimizing. The mean curvature of the minimizers satisfies a first-order system of partial differential equations of “Hodge-type”.

1. Introduction

This paper introduces a geometrically constrained variational problem for the area functional. We consider the area restricted to the lagrangian surfaces of a Kähler surface, or, more generally, a symplectic 4-manifold with suitable metric, and study its critical points and in particular its minimizers. We apply this study to the problem of finding canonical representatives of the lagrangian homology (that part of

The first author was partially supported by NSF grant DMS-9803341. The second author was partially supported by NSF grant DMS-9802487. Parts of this paper were written while the second author was visiting the American Institute of Mathematics in Palo Alto, Ca. Both authors are grateful to AIM for their generous support of this project.

Received January 25, 2001.

the homology generated by lagrangian cycles). We show that the lagrangian homology of a Kähler surface (or of a symplectic 4-manifold) is generated by minimizing lagrangian surfaces that are branched immersions except at finitely many singular points (Theorem 8.1). We precisely describe the structure of these singular points. In particular, these singular points are represented by lagrangian cones with an associated local Maslov index. Only those cones of Maslov index 1 or −1 may be area minimizing (Propositions 7.3 and 7.5). The mean curvature of the minimizers satisfies a first-order system of partial differential equations of “Hodge-type” that we describe precisely below.

We show that an integral homology class may be represented by a lagrangian cycle (a cycle whose two-simplices are given by C^{1} lagrangian maps) if and only the class is annihilated by the symplectic form ω (Proposition 2.1). On the other hand it is known that a homology class may be represented by an immersed lagrangian surface if and only if it is annihilated both by the symplectic form and the first Chern form. For a class which is annihilated by ω, we are able to construct an area minimizing representative among lagrangian competitors and to show that it has a finite number of singularities whose local Maslov classes add to \frac{1}{2}c_{1}(α). In particular, nonflat singularities do arise in minimizing lagrangian surfaces (the note added in proof in [SW] incorrectly states the contrary). In case we have c_{1}(α) = 0, it follows that there are an even number of singular points whose total sum of Maslov indices is 0. We do not know in general whether the minimizer is free of singularities (other than branch points) in this case. We do show however that if a minimizing S^{2} represents a class β ∈ π_{2}(N), and if any integer (> 1) multiple of this S^{2} also minimizes area in its homotopy class, then this S^{2} is free of singularities (other than branch points) (Theorem 8.4).

A situation of great geometric interest in this area is the construction of special lagrangian cycles in Calabi-Yau manifolds (see [SYZ] for the connection with mirror symmetry). The variational method we employ here provides an approach to this construction. In particular, if an immersed lagrangian submanifold Σ of a Kähler-Einstein manifold is stationary for volume, it is automatically minimal (special lagrangian in the Calabi-Yau case) (Lemma 8.2). This may not be true if Σ is singular. In particular, in our two dimensional setting, the surface will be minimal provided there are no nonflat singular points. Note that in the Kähler-Einstein case, it is true that if ω(α) = 0 then c_{1}(α) = 0, and hence the sum of the Maslov indices of the nonflat singular points is zero. We show that the nonflat cones are not minimizing if one allows nonorientable comparison surfaces (Proposition 7.8). This suggests that a lagrangian minimizer in a ℤ_{2} homology class will be minimal, and that the ℤ_{2} homology of any Kähler-Einstein surface (it is easy to show that every ℤ_{2} homology class may be represented by a lagrangian cycle) can be represented by minimal lagrangian cycles. We will study the ℤ_{2} homology in a forthcoming paper.

The technical work in this paper involves constructing area minimizing lagrangian maps from a surface into a symplectic 4 manifold. We follow the variational approach of first constructing such minimizers in the Sobolev class W^{1,2}, and then discussing the regularity of the minimizers. Comparison results of Gromov [G] and Allcock [Al] are used to show the Hölder continuity of such maps (Theorem 2.8). In contrast to the situation for the classical Plateau problem, the higher regularity is a difficult issue. This is because in the classical case the maps are harmonic, while in our case, the minimization of energy among lagrangian maps does not seem to imply any useful regularity. To get such regularity we derive a monotonicity formula (Proposition 3.2), and use it to obtain a partial regularity theorem and the existence of tangent cones (Theorem 4.10). The monotonicity formula is of a rather subtle nature. The setting for this formula is that of contact stationary surfaces in the Heisenberg group. A difficulty which we encounter here in proving the regularity is the issue of controlling both the map and the surface, and as such it involves a combination of classical and Geometric Measure Theory ideas. We believe the methods developed here will be useful in other problems involving the minimization of area among surfaces with geometric constraints (e.g., embeddedness).

We now introduce the technical setting for this problem and describe the PDE which we will be studying. Let N be a symplectic 2n-manifold with symplectic form ω. We will equip (N, ω) with what is sometimes called a compatible metric g and almost complex structure J. That is, the triple (ω, g, J) is required to satisfy, for v, w ∈ T N: ω(v, Jw) = g(v, w), ω(Jv, Jw) = ω(v, w) . An important example of this structure is the case that (N, g, ω, J) is a Kähler manifold with Kähler metric g, Kähler form ω and complex structure J. A vector field X on N that preserves the symplectic form, i.e., that satisfies L_{X}ω = 0 where L denotes the Lie derivative, is called a symplectic vector field. The diffeomorphisms it generates are called symplectic diffeomorphisms. Since L_{X}ω = d(X ω), it follows that X is a symplectic vector field if and only if X ω is closed. If X ω is exact , i.e., X ω = dh, for a function h on N, X is called a hamiltonian vector field for the hamiltonian h. The diffeomorphisms it generates are called hamiltonian diffeomorphisms. Suppose that Σ is a real n-dimensional manifold (compact or with boundary). We say that a smooth (C^{1}) map ℓ : Σ → (N, ω) is lagrangian if ℓ^{∗}ω = 0. If ℓ is an immersion then using the compatible metric the lagrangian condition can be formulated as follows: ℓ(Σ) is lagrangian if at each point x ∈ ℓ(Σ) and for each tangent vector v ∈ T_{x}(ℓ(Σ)) then ⟨Jv, w⟩ = 0 for all w ∈ T_{x}(ℓ(Σ)) (i.e., JT_{x}(ℓ(Σ)) ⊥ T_{x}(ℓ(Σ))). We remark that if Σ ⊂ ℝ^{2n} is given as a graph over the standard real n-plane; i.e., Σ is given as y = f(x) where f is a map from ℝ^{n} to ℝ^{n}, then Σ is lagrangian if and only if there is a function u(x) defined locally such that f = ∇u. This is the simplest way to construct lagrangian submanifolds of ℝ^{2n}.

Lagrangian submanifolds enjoy the following remarkable property. Let Σ be an immersed lagrangian submanifold and suppose that (N, ω) is a Kähler manifold. The mean curvature vector H of Σ in N is a section of the normal bundle on Σ. Define a 1-form on Σ by: σ_{H} = H ω. Then a geometric computation (see the Appendix) gives: dσ_{H} = Ric_{|_{Σ}} where Ric denotes the Ricci 2-form on N. In particular, suppose N is Kähler-Einstein so that the metric satisfies: Ric = Rω where R is the scalar curvature. Then since Σ is lagrangian dσ_{H} = 0 and therefore H is an infinitesimal symplectic motion.

This observation suggests that it is natural to consider variational problems for volume with a lagrangian constraint. Recall that an immersed submanifold is called stationary if the volume is critical for arbitrary smooth compactly supported variations. It is well-known that an immersed submanifold is stationary if and only if it is minimal, i.e., H = 0. We will call an immersed lagrangian submanifold Σ lagrangian stationary if the volume is critical for arbitrary smooth variations through lagrangian submanifolds. We will call Σ hamiltonian stationary if the volume is critical for compactly supported hamiltonian variations. If X is a hamiltonian vector field then X ω = dh or, equivalently X = J∇h. It follows that if Σ is an immersed lagrangian submanifold that is hamiltonian stationary then

0 = ∫_{Σ}⟨H, X⟩ = ∫_{Σ}⟨σ_{H}, dh⟩ = ∫_{Σ}⟨δσ_{H}, h⟩.

Since this holds for any smooth function h with compact support we conclude that δσ_{H} = 0. Combining this with the computation of dσ_{H}, we see that if Σ is an immersed hamiltonian stationary lagrangian submanifold then

(1.1)  dσ_{H} = Ric_{|_{Σ}},
δσ_{H} = 0.

Thus, if N is Kähler-Einstein then σ_{H} is a harmonic 1-form.

The equations (1.1) are third order in Σ. To understand the nature of these equations, we consider the special case in which Σ ⊂ ℝ^{2n} is given by the graph of the gradient of a function u(x). That is, Σ is given by {(x, y = ∇u(x)) : x ∈ ℝ^{n}} for a scalar function u(x). In this case, the volume of Σ is given by ∫ \sqrt{det(g_{ij})} dx where g_{ij} = δ_{ij} + ∑_{k} u_{ik}u_{kj}. Doing a variation of the form u + tv with v of compact support one derives the Euler Lagrange equation

∑_{k=1}^{n} \frac{∂}{∂x_{k}} (Δ_{μ}(u_{k})) = 0

where Δ_{μ} is the Laplace operator with respect to the induced metric μ. In particular we see that this is a quasilinear fourth order elliptic equation whose linearization at u(x) = 0 is the biharmonic equation. We remark that this set of hamiltonian stationary equations has been studied from the point of view of integrable systems by Hélein and Romon [HR].

2. Preliminaries

We will suppose, for the remainder of the paper, that N denotes a symplectic 4-manifold with symplectic form ω, compatible metric g and almost complex structure J.

An integral lagrangian cycle in N is an integral cycle with the 2simplices given by C^{1} lagrangian maps. We say a homology class α ∈ H_{2}(N, ℤ) is a lagrangian homology class if it can be represented by an integral lagrangian cycle.

Proposition 2.1. A homology class α ∈ H_{2}(N, ℤ) is a lagrangian homology class if and only if [ω](α) = 0.

Proof. Suppose that α ∈ H_{2}(N, ℤ) satisfies [ω](α) = 0. Represent α by an embedded surface Σ ⊂ N and let T denote a tubular neighborhood of Σ in N. Since ∫_{Σ} ω = 0, ω is exact on T and there is a one-form η such that on T, ω = dη. Note that η is well-defined up to addition of a closed one-form. Choose a triangulation of Σ that is ε-fine, in the sense that every one-simplex has length less than ε. Let Δ denote a two-simplex of the triangulation. Then

∫_{Δ} ω = ∫_{∂Δ} η,

and the second integral is independent of the choice of η. Let s denote a one-simplex in ∂Δ. Then

|∫_{s} η| < Cε,

where the constant C depends only on η. Perturbing s in T keeping its endpoints fixed we can construct a one-simplex s̃ in T with ∫s̃ η = 0. Doing this for each one-simplex in the triangulation we have a onecomplex in T with the same vertices as the original triangulation and with the property that the integral of η around the perturbation of ∂Δ is zero, for any two-simplex Δ. It follows that the perturbation of ∂Δ spans a lagrangian two-simplex. The result follows. q.e.d.

Note that the requirement [ω](α) = 0 imposes a rationality condition on [ω]. This requirement while weaker than [ω] ∈ H^{2}(N, ℚ) nevertheless implies that for “generic” symplectic structure ω there are no nontrivial lagrangian classes.

Proposition 2.2. Let α ∈ H_{2}(N, ℤ) be a lagrangian homology class. Then α can be represented by a piecewise C^{1} lagrangian map ℓ : Σ → N which is an immersion except at finitely many points {x_{1}, …, x_{k}} ⊂ Σ. At each point x_{j}, j = 1, …, k there is a well-defined local Maslov index m_{x_{j}}. The total Maslov index satisfies:

∑_{j=1}^{k} m_{x_{j}} = \frac{1}{2}c_{1}(N)(α).

Proof. By Proposition 2.1 α can be represented by a lagrangian cycle. The cycle can be chosen so that along the one-simplices the tangent planes of adjacent two-simplices are transverse. We smooth this cycle along the one-simplices outside neighborhoods of the vertices, as follows. Choose a one-simplex s. We can suppose that s lies in a Darboux ball. After a hamiltonian diffeomorphism we can suppose that s is a segment of a coordinate axis. Thus s is the boundary of two smooth lagrangian simplices L_{1} and L_{2}. Subdividing s, if necessary, we can find a lagrangian plane P containing s such that near s, L_{1} is a graph over the half plane H_{1} and L_{2} is a graph over the half plane H_{2}, where P = H_{1} ∪ H_{2}. In particular, for i = 1, 2, in a neighborhood of s, L_{i} is the graph of ∇f_{i}, where f_{i} is a smooth function on H_{i}. Let V be a neighborhood of s in P. Let f be a smooth function on P that coincides with f_{1} on H_{1} \ V and with f_{2} on H_{2} \ V. The required smoothing is effected by gluing in the graph of ∇f. Note that if s must be subdivided to find P then the smoothings can be constructed to agree at the subdivision points.

Let x_{j} be a singular point. Let D_{r}(x_{j}) ⊂ Σ be a disc of radius r, center x_{j}. Choose r so that D_{r}(x_{j}) contains no singular points except x_{j}, so that ℓ(D_{r}(x_{j})) lies in a Darboux neighborhood U and so that ℓ is an immersion on ∂D_{r}(x_{j}). Let D_{j} be an oriented immersed disc in U with ∂D_{j} = ℓ(∂D_{r}(x_{j})) such that the induced orientations agree. The lagrangian immersion ℓ determines a trivialization of T N along Σ \ {x_{1}, …, x_{k}} and in particular along ∂D_{r}(x_{j}). Trivialize T N on D_{j} and compare the two trivializations on ∂D_{j}. The result is an element of π_{1}(U(2)) ≃ ℤ. Choosing a generator of π_{1}(U(2)) associates an even integer, 2m_{x_{j}}, to the point x_{j}. This integer is well-defined independent of the choices of r and D_{j}, provided D_{j} ⊂ U and ℓ is an immersion along ∂D_{r}(x_{j}). We will say that m_{x_{j}} is the local Maslov index of x_{j}

If, at each singular point x_{j}, we replace ℓ(D_{r}(x_{j})) with D_{j} we construct a surface Σ̃, no longer lagrangian, that still represents α. Join x_{1} to x_{2}, x_{2} to x_{3}, …, and x_{k−1} to x_{k} by simple curves on Σ\{x_{1}, …, x_{k}}. Cutting along these curves we can realize Σ̃ as the union of Σ\∪_{j}^{k}D_{r}(x_{j}) and ∪_{j}^{k}D_{j} glued along an S^{1}. To compute c_{1}(N)(Σ̃) we compare the trivializations of T N on the boundaries of Σ \ ∪_{j}^{k}D_{r}(x_{j}) and ∪_{j}^{k}D_{j}. By the previous computation it follows that:

c_{1}(N)(α) = c_{1}(N)(Σ̃) = 2 ∑_{j=1}^{k} m_{x_{j}}.

q.e.d.

The following proposition, due to Gromov and Lees [Le], gives necessary and sufficient conditions for a smooth map to be homotopic to a lagrangian immersion.

Proposition 2.3 (Gromov, Lees). A smooth map ϕ : Σ → N is homotopic to a lagrangian immersion if and only if:

(1) ϕ^{∗}[ω] = 0,

(2) ϕ^{∗}c_{1}(N) = 0,

where [ω] is the cohomology class of ω and c_{1}(N) is the first Chern class of (N, J).

It follows that:

Corollary 2.4. An integral homology class α that satisfies [ω](α) = 0 and c_{1}(N)(α) = 0 can be represented by an immersed lagrangian surface.

Let Σ be a compact surface with Riemannian metric and with volume form dμ. Let ℝ^{n} denote euclidean n-space. We denote the Hilbert space of square integrable maps Σ → ℝ^{n} by L^{2}(Σ, ℝ^{n}) and the Hilbert space of maps Σ → ℝ^{n} with square integrable first derivatives by W^{1,2}(Σ, ℝ^{n}). The energy of a map f ∈ W^{1,2}(Σ, ℝ^{n}) is:

E(f) = \frac{1}{2} ∫_{Σ}⟨df, df⟩dv.

By the Nash imbedding theorem we can suppose that N is isometrically embedded in ℝ^{n}, for some n. Define:

W^{1,2}(Σ, N) = {f ∈ W^{1,2}(Σ, ℝ^{n}) : f(x) ∈ N a.e. x ∈ Σ}.

Recall that closed bounded subsets of W^{1,2}(Σ, N) are weakly closed in W^{1,2}(Σ, N).

We need the following lemma about W^{1,2} maps.

Lemma 2.5. There exists ε_{0} > 0 depending only on N such that if ℓ ∈ W^{1,2}(S^{2}, N) satisfies E(ℓ) ≤ ε_{0}, then ℓ is zero in homology in the sense that ∫_{S^{2}} ℓ^{∗}(γ) = 0 for any smooth closed 2-form γ on N.

Proof. We follow [ScU], and consider smoothing the map ℓ. Think of S^{2} as the standard unit sphere in ℝ^{3}. Let t ∈ S^{2}, and consider a euclidean ball of radius h ≤ 1/2 about t. Consider the averaged map

ℓ^{(h)}(t) = (|S^{2} ∩ B_{h}(t)|)^{−1} ∫_{S^{2}∩B_{h}(t)} ℓ da.

The Poincaré inequality then implies

∫_{S^{2}∩B_{h}(t)} |ℓ − ℓ^{(h)}|^{2} da ≤ ch^{2} ∫_{S^{2}∩B_{h}(t)} |∇ℓ|^{2} da.

It follows that the mean square distance from ℓ^{(h)}(t) to ℓ is small in the ball of radius h. In particular the averaged map ℓ^{(h)} has image which is uniformly close to N independent of h. We may thus project this map into N using the nearest point projection map Π which is smooth in a neighborhood of N. We let ℓ_{h} = Π∘ℓ^{(h)}, and we have that the family of maps ℓ_{h} for 0 < h ≤ 1/2 is a family of smooth maps depending smoothly on h, and from standard results we have that the limit as h → 0 exists in the W^{1,2} norm and is equal to ℓ. Since we assumed that the energy of ℓ is small, it follows that the image of ℓ_{1/2} lies in a small coordinate ball in N, and hence we have that ∫_{S^{2}} ℓ_{1/2}^{∗}γ = 0. Now from the homotopy

formula since γ is a closed form we have \frac{dℓ_{h}^{∗}γ}{dh} = dℓ_{h}^{∗}(\frac{∂ℓ}{∂h} γ). It follows that ∫_{S^{2}} ℓ_{h}^{∗}γ is constant, and therefore 0 for all h ∈ (0, 1/2]. But since the maps ℓ_{h} converge strongly in W^{1,2} to ℓ as h tends to 0, we see that ∫_{S^{2}} ℓ^{∗}γ = 0 as claimed. q.e.d.

The notion of a lagrangian map can be extended to W^{1,2}(Σ, N) as follows: We say a map ℓ ∈ W^{1,2}(Σ, N) is weakly lagrangian if:

ℓ^{∗}ω = 0

almost everywhere on Σ. Note that by Lebesgue point theory, we can equivalently define a map ℓ ∈ W^{1,2}(Σ, N) to be weakly lagrangian if:

∫_{D} ℓ^{∗}ω = 0,

for every disc D ⊂ Σ.

Proposition 2.6. The set of weakly lagrangian maps in W^{1,2}(Σ, N) satisfying a uniform energy bound is closed in the weak topology.

Proof. Let ℓ_{i} be a sequence in W^{1,2}(Σ, N) with E(ℓ_{i}) ≤ c which converges weakly to ℓ ∈ W^{1,2}(Σ, N). We must show that ℓ is lagrangian. Note that any subsequence of ℓ_{i} also converges weakly to ℓ, so we may choose subsequences at will. We first consider the energy measures 1/2|∇ℓ_{i}|^{2}dμ, and using the weak convergence theorem for measures we may assume that these measures converge weakly as measures to a limit measure ν; that is, we assume that for any open set Ω in Σ we have 1/2 ∫_{Ω} |∇ℓ_{i}|^{2}dμ → ∫_{Ω} dν. For any ε_{1} > 0, there is at most a finite set S for which ν({Q}) ≥ ε_{1} for Q ∈ S. If ℓ is not weakly lagrangian, then we can find a Lebesgue point P ∈ Σ \ S such that ℓ^{∗}ω(P) ≠ 0. Thus it follows that for all r sufficiently small we have ∫_{D_{r}(P)} ℓ^{∗}ω ≠ 0.

On the other hand, because P is not in S, we have E_{D_{r}(P)}(ℓ_{i}) ≤ 2ε_{1} for i sufficiently large. We fix such a radius r, and observe that we may choose a radius ρ ∈ [r/2, r] and a subsequence again denoted ℓ_{i} so that ℓ_{i}_{|_{∂Dρ}} has energy bounded by 2r^{−1}ε_{1}. This implies that ℓ_{i}_{|_{∂Dρ}} is continuous, rectifiable and has length bounded by a fixed constant times ε_{1}. It follows that a subsequence converges uniformly to ℓ_{|_{∂Dρ}}. Since the length of ℓ_{i}(∂D_{ρ}) is small we can suppose that the images of ∂D_{ρ} all lie in a Darboux ball B ⊂ N. For each i, ℓ_{i}(∂D_{ρ}) extends to a (not necessarily lagrangian) map ℓ̂_{i} : D_{i} → B with small energy . For ε_{1} sufficiently small (depending only on N) we may apply Lemma 2.5 to show that the cycle ℓ_{i}(D_{ρ}) ∪ ℓ̂_{i}(D_{i}) (which can be represented as a map with small energy from S^{2} to N) is trivial in homology. In B write ω = dη, then we have

0 = ∫_{D_{ρ}} ℓ_{i}^{∗}ω + ∫_{D_{i}} ℓ̂_{i}^{∗}ω = ∫_{D_{i}} ℓ̂_{i}^{∗}ω = ∫_{∂D_{ρ}} ℓ_{i}^{∗}η,

where the last equality follows by Stokes’ theorem (and ℓ̂_{i} = ℓ_{i} on ∂D_{ρ} = ∂D_{i}). Note that the condition that the energy of ℓ_{i}|_{∂D_{ρ}} is bounded implies that there is a subsequence again denoted ℓ_{i} which converges weakly in W^{1,2}(∂D_{ρ}) to ℓ. It follows that ∫_{∂D_{ρ}} ζℓ_{i}^{∗}η converges to ∫_{∂D_{ρ}} ζℓ^{∗}η for any L^{2}(∂D_{ρ}) function ζ. (To see this observe that ℓ_{i}^{∗}η is a linear combination of first derivatives of ℓ_{i} with coefficients which are smooth functions of ℓ_{i}, and these coefficient functions converge in L^{2} norm.) Choosing ζ = 1, we conclude

0 = ∫_{∂D_{ρ}} ℓ^{∗}η.

Repeating the above argument for ℓ, we may find a small energy map ℓ̂: D → B which agrees with ℓ on ∂D = ∂D_{ρ}. As above it follows that

0 = ∫_{D_{ρ}} ℓ^{∗}ω + ∫_{D} ℓ̂^{∗}ω = ∫_{D_{ρ}} ℓ^{∗}ω + ∫_{∂D_{ρ}} ℓ^{∗}η,

where the second equality follows by Stokes Theorem (and ℓ̂ = ℓ on ∂D_{ρ} = ∂D). Therefore we have ∫_{D_{ρ}} ℓ^{∗}ω = 0. This contradiction shows that ℓ is weakly lagrangian. q.e.d.

A classical invariant of lagrangian maps that plays a role in our considerations is the period. We specialize to the case N = ℝ^{4} (with the euclidean metric g and the standard symplectic structure ω = ∑_{i}^{2} dx_{i} ∧ dy_{i}) to describe this invariant. Let ℓ : Σ → ℝ^{4} be a C^{1} lagrangian map and γ : S^{1} → Σ be a closed curve. Choose a 1-form η on ℝ^{4} such that dη = ω. We define the period of γ to be: period(γ) = ∫_{γ} ℓ^{∗}η. This number is well-defined independent of the choice of primitive η and depends only on the homology class of γ in H_{1}(Σ, ℤ). If the periods of all closed curves on Σ vanish we say that the lagrangian map is exact. Equivalently, all periods vanish if and only if ℓ^{∗}η = dφ for a function φ on Σ. There is another description of exact lagrangian maps. Consider ℝ^{5} with coordinates (x, y, φ) and projection p : ℝ^{5} → ℝ^{4}, (x, y, φ) ↦ (x, y). Let α = dφ−(∑_{i} x_{i}dy_{i}−y_{i}dx_{i}) be the contact 1-form on ℝ^{5}. The contact distribution is the distribution of hyperplanes defined by α = 0. A C^{1} map of a surface λ : Σ → ℝ^{5} is called legendrian if λ^{∗}(α) = 0 or, equivalently, if λ(Σ) is everywhere tangent to the contact distribution. A lagrangian map Σ → ℝ^{4} is exact if and only if it admits a legendrian lift Σ → ℝ^{5}. It is easy to construct non-exact lagrangian maps into ℝ^{4}. However every C^{1} lagrangian map ℓ : Σ → N is locally exact in the sense that Σ can be covered by discs {D_{λ}} so that the image of each D_{λ} under ℓ lies in some Darboux ball. The notion of local exactness plays an essential role in our regularity theory. In fact, it is possible to show that any weakly lagrangian map in W^{1,2}(Σ, N) is locally exact in a suitable sense. However for our purposes this will not be necessary.

The following comparison result is due to Gromov ([G]) and Allcock ([Al]). We will use it to prove a Hölder continuity result and (in a later section) a strong compactness theorem for minimizing lagrangian maps.

Proposition 2.7 (Gromov, Allcock). Given a W^{1,2} map φ from the unit circle C into ℝ^{4} satisfying the period condition ∫_{C} φ^{∗}(xdy) = 0, there exists a lagrangian map ℓ in W^{1,2}(D, ℝ^{4}) with ℓ = φ on C, and

A(ℓ(D)) ≤ cL(ℓ(C))^{2}

for a fixed constant c.

We may now use the idea of Morrey ([M1]) to establish the Hölder continuity of minimizing lagrangian maps from a disk. We will consider minimizing lagrangian maps which are weakly conformal in the sense that ∥ℓ_{∗}(∂/∂x)∥ = ∥ℓ_{∗}(∂/∂y)∥ and ⟨ℓ_{∗}(∂/∂x), ℓ_{∗}(∂/∂y)⟩ = 0 a.e.. Such maps will be constructed later in this paper.

Theorem 2.8. Let ℓ ∈ W^{1,2}(D, N) be a weakly conformal, minimizing lagrangian map. There exists α ∈ (0, 1) and c depending only on N such that (after redefinition on a set of measure 0), ℓ is Hölder continuous on D_{1/2} and satisfies the bound

d(ℓ(P), ℓ(Q)) ≤ c(A(ℓ(D)))^{1/2}|P − Q|^{α}

for P, Q ∈ D_{1/2}.

Proof. Consider a point P ∈ D_{1/2} and a radius r < 1/2. We want to show that E_{D_{r}(P)}(ℓ) ≤ c(L(ℓ(∂D_{r}(P)))^{2} for a fixed constant c depending only on N. To see this, observe that if the right hand side is greater than a fixed ε_{0} > 0, the inequality is trivial, so it suffices to assume that L = L(ℓ(∂D_{r}(P)) is small. Thus we may assume that the curve ℓ(∂D_{r}(P)) lies in a Darboux chart in which the metric is nearly euclidean. In this case it follows from Proposition 2.7 that there exists a map ℓ_{0} in W^{1,2}(D_{r}(P), N) with ℓ_{0} = ℓ on ∂D_{r}(P) and A(ℓ_{0}) ≤ cL^{2}. Since ℓ is area minimizing and weakly conformal we have E_{D_{r}(P)}(ℓ) ≤ A(ℓ_{0}) ≤ cL^{2} as required. Now by the Schwarz inequality we have L^{2} ≤ 2πr \frac{d}{dr} E_{D_{r}(P)}(ℓ), and we have shown that E_{D_{r}(P)}(ℓ) ≤ cr \frac{d}{dr} E_{D_{r}(P)}(ℓ). Integrating this differential inequality we get the decay estimate E_{D_{r}(P)}(ℓ) ≤ cr^{2α}E(D) for a fixed α ∈ (0, 1). By Morrey’s Lemma ([GT]) this implies the desired Hölder estimate. q.e.d.

Corollary 2.9. If ℓ ∈ W^{1,2}(Σ, N) is a weakly conformal, minimizing lagrangian map then it is locally exact.

3. The first variation of area

Let Σ be a compact Riemann surface, and ℓ ∈ W^{1,2}(Σ, M) be a weakly conformal, locally exact, lagrangian map. We will assume throughout this section that there is a disk D in Σ such that ℓ(D) is contained in a fixed Darboux chart O in N. Note that by Theorem 2.8, this will be true on sufficiently small disks if the map is minimizing. We will also assume that ℓ is stationary in the sense that the first variation of area is zero for allowable deformations of ℓ through lagrangian maps. We first define the variations which we will consider. We suppose that x, y are local Darboux coordinates in O, so that the symplectic form is in the standard form ω = dx ∧ dy. There is an obvious class of variations given in terms of compactly supported hamiltonian functions h(x, y), namely the flow of the vector field X_{h} = h_{x} \frac{∂}{∂y} − h_{y} \frac{∂}{∂x}. If F_{t} denotes this flow, then ℓ_{t} = F_{t} ∘ ℓ is an allowable variation. It will be important to extend this class of variations to allow variations which include dilations, and other collapsing deformations. In order to do this we use the local exactness. This enables us to define a function φ on ℓ^{−1}(O) which satisfies dφ = ℓ^{∗}η where η = (xdy − ydx). We wish to consider hamiltonians which depend on φ as well as x, y. In order to do this, we may form the 5 dimensional space with coordinates x, y, φ. In this space we have the contact 1-form α = dφ − η, and we may consider the contact transformations. These are generated by vector fields, again denoted X = X_{h}, which depend on a hamiltonian h(x, y, φ). They are given by

X_{h} = h_{x} \frac{∂}{∂y} − h_{y} \frac{∂}{∂x} − h_{φ} (x\frac{∂}{∂x} + y \frac{∂}{∂y} ) + (−2h + (xh_{x} + yh_{y}))\frac{∂}{∂φ}.

One can check that the above vector field satisfies

L_{X_{h}}α = −2h_{φ}α,

and hence its flow preserves the contact distribution. To construct a variation generated by such a vector field, we may take its flow F_{t} in ℝ^{5}. This flow preserves the contact distribution defined by α. If we denote by ℓ̃the local lift of ℓ given by ℓ̃= (ℓ, φ), then we may construct a variation by setting ℓ_{t} = Π∘F_{t}∘ℓ̃where Π(x, y, φ) = (x, y) is the natural projection map. The first variation of the area for such a variation is given by

\frac{d}{dt}A(ℓ_{t})|_{t=0} = ∫_{Σ} div_{Σ}(X_{h}) da = 0 (3.1)

where we have div_{Σ}(X) = ∑_{i=1}^{2}⟨∇_{e_{i}}X, e_{i}⟩ with e_{1}, e_{2} being an orthonormal basis for T ℓ(Σ) at a point and ∇ being the lift under Π of the Levi-Civita connection from ℝ^{4} to ℝ^{5}. We have also lifted the metric g by pullback under Π to a (degenerate) metric on ℝ^{5}. In order to use (3.1) effectively, we will first need to do some calculations for the Euclidean metric, and then we will use the fact that g can be made nearly Euclidean near any given point by a suitable choice of coordinates.

We let ∇^{0} denote the Euclidean connection, and let div_{0}(X) denote the corresponding operator with respect to the Euclidean metric (denoted as the dot product) and connection; thus we have div_{0}(X) = ∑_{i=1}^{2} ∇_{e_{i}}^{0}X · e_{i} where e_{1}, e_{2} is a Euclidean orthonormal basis. In particular, we see that div_{0}(X) does not depend on the \frac{∂}{∂φ} component of X since the inner product does not involve φ. On the other hand, the e_{j} in general have a \frac{∂}{∂φ} component, so the φ dependence of the first 4 components of X is significant. We define an important function s by s = \frac{1}{2}(x^{2} + y^{2}). Since we are interested in the first variation, and it only depends on the first 4 components of a contact vector field, we will use the notation X ≈ Y to mean that X and Y have the same first 4 components. The contact vector fields associated with s, φ then satisfy

X_{s} ≈ x\frac{∂}{∂y} − y \frac{∂}{∂x} (3.2)

and

X_{φ} ≈ −x\frac{∂}{∂x} − y \frac{∂}{∂y} ≈ J_{0}X_{s} (3.3)

where J_{0} is the standard complex structure in the x, y coordinates. It turns out to be very useful to do a change of variables using the fact that s is positive. We introduce t + iθ = log(s + iφ) where we choose θ ∈ [−\frac{π}{2} , \frac{π}{2} ]. In other words, we set s̃ = \sqrt{s^{2} + φ^{2}}, t = log(s̃), and θ = tan^{−1}(φ/s). The function s̃ will play the role of the square of the distance function. In particular, if we set r_{0} = \sqrt{2}(s^{2} + φ^{2})^{1/4} so that s̃ = \frac{1}{2}r_{0}^{2}, then the function r_{0} will play the role of the distance to the origin. We define balls B_{σ}(0) to be the points p with r_{0}(p) < σ. (We note that the function r_{0} arises naturally in the function theory of the sub-laplacian on the Heisenberg group; in fact, the fundamental solution of this operator with pole at 0 is a function of r_{0}.) In order to derive monotonicity identities, we consider functions η(t, θ) with compact support, and use the vector field X_{η} in the first variation formula (3.1). One can check the following calculations

div_{0}(X_{θ}) = −2|∇^{T} θ|^{2}
div_{0}(X_{t}) = −2\frac{sin θ}{s̃} − 2∇^{T} θ · ∇^{T} t.

Since X_{η} ≈ η_{t}X_{t} + η_{θ}X_{θ}, we can compute

div_{0}X_{η(t,θ)} = η_{θt}(|∇^{T} θ|^{2} − |∇^{T} t|^{2}) − 2η_{t}s^{−1} sin θ − 2η_{θ}|∇^{T} θ|^{2}
+ (η_{tt} − η_{θθ} − 2η_{t})∇^{T} θ · ∇^{T} t.

Using the fact that |∇^{T} t|^{2} + |∇^{T} θ|^{2} = 2s^{−1} cos θ, we may rewrite this

div_{0}X_{η(t,θ)} = (2η_{θt} − 2η_{θ})|∇^{T} θ|^{2} − 2η_{θt}s^{−1} cos θ − 2η_{t}s^{−1} sin θ
+ (η_{tt} − η_{θθ} − 2η_{t})∇^{T} θ · ∇^{T} t.

We now assume that we have chosen Darboux coordinates centered at a given point P such that the metric g agrees with the Euclidean metric at P. Since the connection terms are bounded and the metrics agree at P, we may express the first variation condition (3.1) as

∫_{Σ} div_{0}Xda_{0} = R(X) (3.4)

where the remainder term satisfies

|R(X)| ≤ c ∫_{Σ}(\sqrt{s}|∇X| + |X|)da. (3.5)

We will choose a function ζ(t) below with the properties that it is a smooth nonincreasing function which is identically zero for t ≥ log(1/2) and identically equal to 1 − 2λe^{t} for t ≤ −c for positive constants c and λ (chosen below). We then define ψ by ψ(t) = −1/2e^{−t}ζ′(t), and we see that

∫_{−∞}^{∞} e^{t}ψ(t) dt = 1/2. (3.6)

We thus have ζ(t) = 1 − 2 ∫_{−∞}^{t} e^{τ} ψ(τ) dτ. For a radius a > 0, we let ψ_{a}(t) = ψ(t − 2 log(a)) and ζ_{a}(t) = ζ(t − 2 log(a)). We then let η(t, θ) be the solution of the wave equation η_{tt} − η_{θθ} − 2η_{t} = 0 with initial data η(t, 0) = 0, \frac{∂η}{∂θ} (t, 0) = ζ(t). (Note that θ is the time variable in the wave equation and t is the space variable.) We then set η_{a}(t, θ) = η(t − 2 log(a), θ). Since the range on θ is between −π/2 and π/2, we see from domain of dependence considerations that

η_{a}(t, θ) = θ − 2λ\frac{e^{t}}{a^{2}} sin θ

for t < log(a^{2}) − c − π/2 and η_{a}(t, θ) = 0 for t > log(a^{2}/2) + π/2. We observe furthermore that the functions F_{a}(t, θ), G_{a}(t, θ) given by

G = η_{θ} − η_{θt}, G_{a}(t, θ) = G(t − 2 log(a), θ)

and

F = −1/2(η_{θt}s^{−1} cos θ + η_{t}s^{−1} sin θ)
= −1/2e^{−t}(η_{θt} cos θ + η_{t} sin θ),
F_{a}(t, θ) = F (t − 2 log(a), θ)

satisfy F = G = 0 for t > log(a^{2}/2) + π/2 and F = λ, G = 1 for t < log(a^{2}) − c − π/2. We also note that the initial value of F_{a}, given by F_{a}(t, 0) = ψ_{a}(t), is specified with the constraints above. Thus we have

div_{0}X_{η_{a}} = −2G_{a}|∇^{T} θ|^{2} + \frac{4}{a^{2}} F_{a}.

We now choose η = η_{a} − η_{b} where 0 < b < a, and consider the first variation of area with vector field X_{η}. This gives by (3.4)

a^{−2} ∫_{Σ} F_{a}da_{0} − b^{−2} ∫_{Σ} F_{b}da_{0} = \frac{1}{2} ∫_{Σ}(G_{a} − G_{b})|∇^{T} θ|^{2}da_{0} + R(X_{η}). (3.7)

We now describe the choice of ζ. Let c be a positive number such that c + log(1/2) > 2πe^{π/2} and then set τ = \frac{1}{4}(c + log(1/2)). Note that τ > π/2. Let α be a nonincreasing, nonnegative function constructed as follows. Define a piecewise linear function β by:

β(t) = \{ \begin{matrix}1 & t < −c \\ −\frac{1}{4τ} (t − log(1/2)) & −c ≤ t ≤ log(1/2) \\ 0 & t > log(1/2). \end{matrix}

Choose t_{0} = −c + 2τ so that β(t_{0}) = 1/2. Then define α to be β with the corners at t = −c and t = log(1/2) smoothed and such that:

(i) 0 ≤ α ≤ 1.

(ii) α′ ≤ 0.

(iii) α′′(t) ≤ 0 for t < t_{0} + τ.

(iv) α′′(t) ≥ 0 for t > t_{0} − τ.

(v) α has the symmetry α(t) = 1 − α(2t_{0} − t).

Finally, let ζ be the solution of the ODE ζ′ − ζ = −α with ζ(t) = 0 for t > log(1/2). Then it follows that ζ is a smooth nonincreasing function which is identically zero for t > log(1/2) and equal to 1 − λe^{t} for t < −c and some positive number λ.

The following result summarizes the conditions we will need concerning the functions F and G.

Proposition 3.1. With the choices above, the function F is nonnegative, and G satisfies 0 ≤ G ≤ 1 for −π/2 ≤ θ ≤ π/2. Furthermore, there is a fixed number θ_{0} ∈ (0, 1) such that G_{a}−G_{b} ≥ 0 for 0 < b ≤ θ_{0}a.

Proof. The transformation u = e^{−t}η changes the initial value problem for η into the following initial value problem for u:

u_{tt} − u_{θθ} − u = 0
u(t, 0) = 0
u_{θ}(t, 0) = e^{−t}ζ(t).

Utilizing the Riemann function it is possible to explicitly solve this initial value problem [CH]. Let J_{0}(x) denote the Bessel function of the first kind of order zero. Then the solution is u = e^{−t}η where

η(t, θ) = \frac{1}{2} ∫_{−θ}^{θ} J_{0} (\sqrt{(θ − μ)(θ + μ)}) e^{−μ}ζ(μ + t)dμ (3.8)

where J_{0}(σ) is the solution of the ODE

J′′_{0} + \frac{1}{σ} J′_{0} + J_{0} = 0, J_{0}(0) = 1. (3.9)

The basic properties of J_{0} which we shall need are that J_{0}(σ) > 0, J′_{0}(σ) < 0, and J′′_{0}(σ) < 0 for 0 ≤ σ ≤ π/2. The fact that J′′_{0} and J′_{0} are negative up to the first zero of J_{0} is shown in [CH, p. 495]. It follows that J′′_{0} +J_{0} > 0 for 0 < σ < σ_{0} where σ_{0} is the first zero of J_{0}. It remains to show that σ_{0} ≥ π/2. To see this, we set v_{ε}(σ) = J_{0}(σ)/ cos(ε+σ) for 0 ≤ σ < π/2 − ε, and we compute J′′_{0}(σ) = v′′_{ε}(σ) cos(ε + σ) − 2v′_{ε}(σ) sin(ε + σ) − J_{0}(σ). It follows that v′′_{ε}(σ) cos(ε + σ) > 2v′_{ε}(σ) sin(ε + σ) for 0 ≤ σ < min{σ_{0}, π/2 − ε}. Thus the function v_{ε} can not achieve a local maximum in (0, min{σ_{0}, π/2 − ε}). On the other hand we have v′_{ε}(0) = sin(ε)/ cos^{2}(ε) > 0 for ε > 0, so it follows that v′_{ε}(σ) > 0 for σ small. Therefore we see that v_{ε} is an increasing function for 0 ≤ σ < min{σ_{0}, π/2 − ε} for any ε > 0. In particular it follows that σ_{0} ≥ π/2 and setting v = lim_{ε→0} v_{ε} we have v(σ) ≥ cos(σ) on this interval.

To verify the proposition, we first need only check that F is nonnegative. We see from (3.8) that

η_{t}(t, θ) = \frac{1}{2} ∫_{−θ}^{θ} J_{0} (\sqrt{θ^{2} − μ^{2}}) e^{−μ}ζ′(μ + t)dμ.

Differentiating in θ we have

η_{tθ} = \frac{1}{2}\{e^{−θ}ζ′(t + θ) + e^{θ}ζ′(t − θ)

+ ∫_{−θ}^{θ} \frac{∂}{∂θ} J_{0} (\sqrt{θ^{2} − μ^{2}}) e^{−μ}ζ′(μ + t)dμ\}.

Substituting ζ′ = −2e^{t}ψ, we then have the explicit formula for F

F (t, θ) = \frac{1}{2}(ψ(t + θ) + ψ(t − θ)) cos(θ)

+\frac{1}{2} ∫_{−θ}^{θ} \{ sin(θ)J_{0} (\sqrt{θ^{2} − μ^{2}})

+ cos(θ)\frac{∂}{∂θ} J_{0} (\sqrt{θ^{2} − μ^{2}}) \}ψ(μ + t)dμ.

Now we observe that the integrand above is nonnegative for 0 < θ < π/2. To see this, note that it suffices to show that the function Q(θ) = J_{0}(\sqrt{θ^{2} − μ^{2}})/ cos(θ) is increasing in this range. Recall that we showed above that the function v(σ) = J_{0}(σ)/ cos(σ) is increasing for 0 < σ < π/2, and note that we have

Q(θ) = v(\sqrt{θ^{2} − μ^{2}})\frac{cos(\sqrt{θ^{2} − μ^{2}})}{cos(θ)},

and it is elementary to check that the ratio of cosine functions is increasing in θ for 0 < θ < π/2 and |μ| ≤ θ. We have therefore shown that F ≥ 0.

To analyze G we begin by showing that G(t, θ) ≥ 0. We first observe that u = e^{−t}G is a solution of the wave equation u_{tt} − u_{θθ} − u = 0 with initial data given by u(t, 0) = u_{0}(t) = e^{−t}(ζ − ζ′) = e^{−t}α and u_{θ}(t, 0) = 0. We then have the explicit formula

(3.10)  u(t, θ) = \frac{1}{2}\{u_{0}(t + θ) + u_{0}(t − θ)

+ ∫_{−θ}^{θ} \frac{∂}{∂θ} J_{0}(\sqrt{θ^{2} − μ^{2}})u_{0}(t + μ) dμ\}.

We note that we have

cos(θ) = 1 + \frac{1}{2} ∫_{−θ}^{θ} \frac{∂}{∂θ} J_{0}(\sqrt{θ^{2} − μ^{2}}) dμ

since both sides represent the solution of the Cauchy problem v_{tt} −v_{θθ} − v = 0, v(t, 0) = 1, v_{θ}(t, 0) = 0. Since u_{0} = e^{−t}α and α′′ ≥ 0 for t ≥ t_{0}−τ, we have u′′_{0} = e^{−t}(α′′ − 2α′ + α) ≥ 0 for t ≥ t_{0} − τ. It follows that the function u_{0} is a convex function for t ≥ t_{0} − τ. Suppose t ≥ t_{0} and |θ| ≤ π/2. If ℓ(t+μ) denotes the linear function with ℓ(t+θ) = u_{0}(t+θ) and ℓ(t − θ) = u_{0}(t − θ), then we have u_{0}(t + μ) ≤ ℓ(t + μ) for |μ| ≤ |θ|. (Recall that τ > π/2.) Hence, for t ≥ t_{0} and |θ| ≤ π/2, we have

G(t, θ) ≥ \frac{1}{2}e^{t}\{u_{0}(t + θ) + u_{0}(t − θ)

+ ∫_{−θ}^{θ} \frac{∂}{∂θ} J_{0}(\sqrt{θ^{2} − μ^{2}})ℓ(t + μ) dμ\}.

Since \frac{∂}{∂θ} J_{0}(\sqrt{θ^{2} − μ^{2}}) is an even function of μ whose integral is greater than or equal to −2, it follows that the right hand side is nonnegative, and so G(t, θ) ≥ 0 for t ≥ t_{0}. We next consider the case t ≤ t_{0}. From (3.10), using that α is nonincreasing and \frac{∂}{∂θ} J_{0}(\sqrt{θ^{2} − μ^{2}}) ≤ 0, we have:

(3.11)  G(t, θ) ≥ \frac{1}{2}\{e^{−θ}α(t + θ) + e^{θ}α(t − θ)

+ α(t − θ) ∫_{−θ}^{θ} \frac{∂}{∂θ} J_{0}(\sqrt{θ^{2} − μ^{2}})e^{−μ} dμ\},

Note that u(t, θ) = e^{−t} is the unique solution of the Cauchy problem: u_{tt}−u_{θθ}−u = 0, with initial data given by u(t, 0) = e^{−t} and u_{θ}(t, 0) = 0. From the explicit formula (3.10) for the solution with this initial data we conclude:

1 = \frac{1}{2} \{e^{−θ} + e^{θ} + ∫_{−θ}^{θ} \frac{∂}{∂θ} J_{0}(\sqrt{θ^{2} − μ^{2}})e^{−μ} dμ\} .

Using this in (3.11) we have:

G(t, θ) ≥ α(t − θ) + \frac{1}{2}e^{−θ}(α(t + θ) − α(t − θ)).

If t ≤ t_{0} and |θ| ≤ π/2 this implies:

G(t, θ) ≥ α(t + π/2) − \frac{1}{2}e^{π/2} \frac{π}{c + log(1/2)}
≥ α(t + π/2) − \frac{1}{4} > 0,

since α(t + π/2) > α(t + τ) ≥ \frac{1}{4}. Hence we conclude that G(t, θ) ≥ 0 for all t and |θ| ≤ π/2.

To see that G ≤ 1, we observe that the function v = e^{−t}(1 − G) satisfies the same wave equation as u, and has initial conditions v(t, 0) = v_{0}(t) = e^{−t}(1 − α(t)) = e^{−t}α(2t_{0} − t) and v_{θ}(t, 0) = 0. We can then apply the previous argument to conclude that (1 − G(t, θ)) ≥ 0. q.e.d.

We have shown the (approximate) monotonicity of the quantity a^{−2} ∫_{Σ} F_{a} as a function of the radius a. Observe that because of the normalization (3.6), we have lim_{a→0}(πa^{2})^{−1} ∫_{Σ} F_{a} = 1 at a smooth point of Σ.

Proposition 3.2. Let ℓ : D → N be a weakly lagrangian, weakly conformal, contact stationary map having image in a Darboux chart (centered at ℓ(0)). There exist positive constants c_{1}, c_{2} depending only on N and the area of ℓ such that for any σ ∈ (0, 1) we have

c_{1} ≤ σ^{−2}A(ℓ(Σ) ∩ B_{σ}(0)) ≤ c_{2}.

Proof. We choose a = 1 in (3.7), and observe that by (3.5) we have

|R(X_{η})| ≤ |R(X_{η_{1}})| + |R(X_{η_{b}})| ≤ c + cb^{−1}A(ℓ(Σ) ∩ B_{λb})

where λ > 1 is fixed. From (3.7), we then have for any b ∈ (0, θ_{0})

b^{−2}A(λ^{−1}b) ≤ c + cb^{−1}A(λb).

This implies that

sup_{t∈(0,s)} t^{−2}A(t) ≤ c + cs sup_{t∈(0,λ^{2}s)} t^{−2}A(t)

for any s ∈ (0, λ^{−1}θ_{0}). Fixing s such that cs < 1/2, we then get the bound

sup_{t∈(0,s)} t^{−2}A(t) ≤ c + sup_{t∈(s,1)} t^{−2}A(t) ≤ c

which is the desired upper bound. From the upper bound and (3.7) it follows that for each P ∈ Σ and σ > 0 we have

sup_{t∈(0,θ_{0}σ)}(πt^{2})^{−1} ∫_{Σ} F_{t}da ≤ (πσ^{2})^{−1} ∫_{Σ} F_{σ}da + cσ.

It follows that the limit

Θ(P) = lim_{σ→0}(πσ^{2})^{−1} ∫_{Σ} F_{σ}da

exists and is upper semicontinuous. At any Lebesgue point P of the derivative of ℓ at which the derivative is nonzero, we have Θ(ℓ(P)) ≥ 1. Since such points have full measure with respect to the induced measure da, we have Θ ≥ 1 at all points of the support of Σ. The lower bound now follows. q.e.d.

The above result implies the following important continuity estimate which is an improvement on Theorem 2.8. In the following results, the notation ℓ̃ is used to denote the legendrian lift of the exact lagrangian map ℓ. Note that A(ℓ) = A(ℓ̃), but that the balls we consider lie in ℝ^{5}, and hence the monotonicity formula exists there also.

Proposition 3.3. Suppose that ℓ is a weakly conformal, exact, lagrangian stationary map from the unit disk D_{1} into N. Assume further that there exists r_{0} > 0 and a constant c such that for r ≤ r_{0} we have A(ℓ̃(D_{1})∩B_{r}(P)) ≤ cr^{2} for any P ∈ N. Then ℓ̃is Hölder continuous in D_{1/2}. Moreover, there is a constant ε_{0} depending only on N such that if A(ℓ(D_{1})) ≤ ε_{0}, then there is a uniform bound on the Hölder modulus of continuity of ℓ̃ in D_{1/2}.

Proof. It suffices to prove the last statement, since we may cover D_{1/2} with small balls of small area and apply the last statement to a rescaled version of ℓ̃ to prove the Hölder continuity. To prove the last statement, we may rescale the map in the image so that A(ℓ̃(D_{1})) = 1, and this rescaling only flattens the metric on the image. We now claim that there is a constant θ < 1 such that A(ℓ̃(D_{1/2})) ≤ θ. To prove this, we need only show that there is a positive constant δ such that A(ℓ̃(D_{1} \ D_{1/2})) ≥ δ for then our conclusion follows with θ = 1 − δ. Suppose to the contrary that A(ℓ̃(D_{1} \ D_{1/2})) is small. We can then find a radius σ ∈ [1/2, 1] such that the length L(ℓ̃(∂D_{σ})) is small, say less than ε. Let P be a point of the curve ℓ̃(∂D_{σ}), and choose Darboux coordinates centered at P in which the metric is Euclidean at the origin. Construct a smooth function ζ(r) such that ζ(r) = 0 for r ≤ ε, ζ(r) = 1 for r ≥ 2ε, and ε|ζ′(r)|+ε^{2}|ζ′′(r)| ≤ c. We then consider the hamiltonian h = −φ · ζ ∘ r_{0}. The corresponding contact vector field satisfies X ≈ ζ ∘ r_{0}(x\frac{∂}{∂x} + y \frac{∂}{∂y}) − φX_{ζ∘r_{0}}. Using this vector field in the first variation formula and using easy bounds we get:

A(ℓ̃(D_{σ})\B_{2ε}) ≤ c∫_{D_{σ}∩ℓ̃^{−1}(B_{2ε}\B_{ε})}(|Dφ||Dζ ∘ r_{0}|+|φ|| div_{Σ}(X_{ζ∘r_{0}})|) dμ
≤ c ∫_{D_{σ}∩ℓ̃^{−1}(B_{2ε}\B_{ε})}(ε · ε^{−1} + ε^{2} · ε^{−2}) dμ
≤ cA(ℓ̃(D_{σ}) ∩ B_{2ε}) ≤ cε^{2}.

This implies A(ℓ̃(D_{σ})) ≤ cε^{2} which is a contradiction if ε is small. This verifies our claim that A(ℓ̃(D_{1/2})) ≤ θ.

We complete the proof by observing that we can iterate this inequality to obtain A(ℓ̃(D_{2^{−n}})) ≤ θ^{n} which implies that A(ℓ̃(D_{r})) ≤ cr^{2α} for some α > 0. This same inequality then holds for disks with any center point in D_{1/2}, and by Morrey’s lemma (since A(ℓ̃) is the Dirichlet integral of ℓ) this implies that ℓ has a uniform Hölder estimate on D_{1/2}. Since |Dφ| ≤ c, and the Dirichlet integral is conformally invariant, we see that ∫_{D_{r}} |∇φ|^{2} dt ≤ cr^{2α}, and φ is therefore also Hölder continuous.

q.e.d.

We will need the following global Hölder estimate under the condition that the restriction of ℓ to ∂D_{1} has finite energy.

Proposition 3.4. Suppose that ℓ is a weakly conformal, exact, lagrangian stationary map from the unit disk D_{1} into N. Assume that there exists r_{0} > 0 and a constant c such that for r ≤ r_{0} we have A(ℓ̃(D_{1}) ∩ B_{r}(P)) ≤ cr^{2} for any P ∈ N. Finally assume that the trace of ℓ as a map from the unit circle to N has energy bounded by a constant c. Then ℓ̃ is Hölder continuous on all of D_{1}. Moreover, there is a constant ε_{0} depending only on N such that if A(ℓ(D_{1})) ≤ ε_{0}, then there is a uniform bound on the Hölder modulus of continuity of ℓ̃ on D_{1}.

Proof. The proof is similar to the previous one. We show that

∫_{D_{r}(t)∩D_{1}}(|∇ℓ|^{2} + |∇φ|^{2}) dt ≤ cr^{2α}

for all t ∈ D_{1}. If a disk D_{r}(t) intersects ∂D_{1} in a point t_{0}, we may instead consider the larger disk D_{2r}(t_{0}) and prove the bound there. Thus it suffices to consider disks which are either completely contained in D_{1}, or centered on ∂D_{1}. The first case is done above. To handle disks centered on ∂D_{1}, we need to use the fact that ℓ has a C^{0,1/2} estimate on ∂D_{1} which follows from the bound on the boundary energy. With the aid of this bound, the proof proceeds as above. q.e.d.

We close this section by analyzing the situation for which the monotonicity inequality is an equality.

Proposition 3.5. Let ℓ be a weakly conformal, exact, lagrangian stationary map from Ω ⊂ ℝ^{2} into ℝ^{4} with euclidean metrics. Suppose that ℓ(0) = 0, that ℓ^{−1}(B_{a}) is compact and a^{−2} ∫_{Ω} F_{a} ∘ ℓ da is constant for a ≤ a_{0}. It follows that C = ℓ̃(Ω) ∩ B_{a_{0}} is contained in {φ = 0}, and is a cone over 0 in the sense that for almost every t ∈ ℓ^{−1}(C), the position vector (x, y) = ℓ(t) ∈ C, is tangent to C. If the map ℓ is a smooth immersion in a neighborhood U of t ∈ ℓ^{−1}(C), then ℓ(U) is a portion of a cone.

Proof. From (3.7), we see that θ is a constant function if equality holds in the monotonicity inequality. This means that φ = as for some constant a. We want to show that a = 0, so we note that r_{0}^{2} = 2(1 + a^{2})s, so we choose a hamiltonian function ζ ∘ s where ζ is a monotone increasing, concave function with ζ(0) = 0 and ζ = 1 outside B_{a_{0}}. Using the first variation formula and the fact that div(X_{s}) = 0, we find ∫ ζ′′(s)Ds·X_{s} dμ = 0. If a ≠ 0, this implies that ∫ ζ′′(s)Dφ·X_{s} dμ = 0. On the other hand, Dφ · X_{s} = |Dφ|^{2}, so since ζ′′ > 0 we see that φ is a constant function. Since φ(0) = 0, we have shown that φ is identically zero and hence a = 0 as claimed. The condition that Dφ = 0 implies that the position vector is tangent to the image of ℓ, and this condition easily implies that any regular piece of the image is a cone over 0. q.e.d.

4. Regularity for the two dimensional mapping problem

In this section we discuss the regularity properties of maps ℓ from a surface to a symplectic 4-manifold (N, ω) with almost complex structure J and compatible Riemannian metric g. Since the theorem is local, we assume that ℓ is defined on the unit disk D in the plane with coordinates (t^{1}, t^{2}). We use the notation D_{r}(p) to denote the disk of radius r centered at p ∈ ℝ^{2} with D_{r} denoting the ball with center at the origin. We make the following assumptions on the map where we use the notation ℓ_{i} to denote the weak partial derivative ℓ_{t^{i}}:

i) The map ℓ is a W^{1,2} map from D to N which is weakly conformal and weakly lagrangian in the sense that for almost every (t^{1}, t^{2}) ∈ D we have ∥ℓ_{1}∥^{2} = ∥ℓ_{2}∥^{2}, ⟨ℓ_{1}, ℓ_{2}⟩ = 0, and ⟨Jℓ_{1}, ℓ_{2}⟩ = 0.

ii) The map ℓ is exact on D.

iii) The restriction of ℓ to any subdisk is stationary as a contact map in the sense that for any P ∈ D and r ∈ (0, 1 − |P |], and any smooth contact vector field X which vanishes in a neighborhood of ℓ(∂D_{r}(P)) we have

\frac{d}{dt}A(ℓ_{t}(D_{r}(P)))|_{t=0} = 0 (4.1)

where ℓ_{t} = F_{t} ∘ ℓ and F_{t} is the flow generated by X.

Note that the restriction of ℓ to ∂D_{r}(P) is a W^{1,2} map for almost every r ∈ (0, 1 − |P |], and hence this restriction is continuous, and the length of the curve ℓ(∂D_{r}) is finite. It is thus clear that condition (iii) allows for many variations. Using (ii), (iii) and monotonicity it follows (Theorem 2.8 and Proposition 3.3) that ℓ is continuous. We can thus suppose that for small r the image of ℓ lies in a Darboux neighborhood U. Let (x, y) be Darboux coordinates in U centered at a point with coordinates (0, 0) such that g, J are standard at (0, 0). Let Ũ denote the contact lifting of U. Then ℓ lifts to a legendrian map (again denoted ℓ) from D_{r} → Ũ. With respect to the contact coordinates on Ũ we can write ℓ = (x, y, φ). On D_{r} we have:

dφ = ∑_{i}(x_{i}dy_{i} − y_{i}dx_{i}).

This expresses the exactness of ℓ on D_{r}. We state the following regularity result.

Theorem 4.1. There exists ε_{0} > 0 such that if there is a conformal linear lagrangian map ℓ_{0} with ℓ_{0}(0) = 0 and an r < 1/2 for which:

r^{−2} ∫_{D_{r}} |∇(ℓ − ℓ_{0})|^{2} dt + r^{−4} ∫_{D_{r}} |ℓ − ℓ_{0}|^{2}dt < ε_{0},
r^{−2} sup_{t∈D_{r}} |ℓ_{0}(t) − (t, 0)|^{2} < ε_{0},
∥g − δ∥_{C^{2}(B_{1})} < ε_{0}

then ℓ is a smooth embedding on D_{r/2}.

The following result is necessary for controlling the parametrization of maps constructed as rescaled limits of ℓ.

Proposition 4.2. Assume that there is a constant c_{1} such that A(ℓ(D_{1})) ≤ c_{1}, and inf_{t∈D_{1/4}} A(ℓ(D_{1/8}(t))) ≥ c_{1}^{−1}A(ℓ(D_{1})). There is a constant c depending only on c_{1} such that for all t ∈ D_{1/4} and all r ≤ 1/4 we have

A(ℓ(D_{r}(t))) ≥ c^{−1}A(ℓ(D_{2r}(t))).

Proof. Fix a point t_{0} ∈ D_{1/4}, and let A(r) = A(ℓ(D_{r}(t_{0}))) which is a continuous, monotone increasing function. The derivative A′(r) then exists for almost every r and is equal to ∫_{∂D_{r}(t_{0})} |∇ℓ|^{2} ds. By differentiation theory of monotone functions we have

∫_{σ}^{2σ} \frac{A′(r)}{A(r)} dr ≤ log (\frac{A(2σ)}{A(σ)}) . (4.2)

Now if we choose r ∈ (0, 1/4], we may find a unique nonnegative integer k such that r = 2^{−k}r_{0} with r_{0} ∈ (1/8, 1/4]. We let r_{j} = 2^{−j}r_{0} for j = 0, 1, …, k so that r = r_{k}. We will prove that A(r_{j}) ≥ c^{−1}A(2r_{j}) for a fixed constant c by induction on j. We begin with j = 0, and observe that the inequality follows with c = c_{1} by hypothesis since 1/8 < r_{0} and 2r_{0} ≤ 1/2 so that D_{2r_{0}}(t) ⊂ D_{1}. We now prove the inductive step. Assume that A(r_{j−1}) ≥ c^{−1}A(2r_{j−1}) for some j ≥ 1. Applying the mean value theorem with σ = r_{j−1} in (4.2) and using the inductive assumption, we find σ_{0} ∈ [1, 2] with r_{j−1}A′(σ_{0}r_{j−1}) ≤ log(c)A(σ_{0}r_{j−1}). We now rescale the map ℓ by setting ℓ(τ) = (A(σ_{0}r_{j−1}))^{−1/2}ℓ(σ_{0}r_{j−1}τ + t_{0}). We then define A(r) by

A(r) = A(ℓ(D_{r})) = \frac{A(σ_{0}r_{j−1}r)}{A(σ_{0}r_{j−1})}.

Our conditions then imply A(1) = 1 and A′(1) ≤ log(c), and we need to show that A(2^{−1}σ_{0}^{−1}) ≥ c^{−1}A(σ_{0}^{−1}). Because σ_{0} ∈ [1, 2], it suffices to show that A(1/4) ≥ c^{−1}. By the mean value inequality we may choose σ_{1} ∈ [1/8, 1/4] such that A′(σ_{1}) ≤ 8A(1/4). If we set δ = (A(1/4))^{1/2}, we may apply the Schwarz inequality to show that

L(ℓ(∂D_{σ_{1}})) ≤ (2πσ_{1})^{1/2}(A′(σ_{1}))^{1/2} ≤ 2π^{1/2}δ.

Let ρ(P) for P ∈ N be the (modified) distance function from a point P_{0} ∈ ℓ(∂D_{σ_{1}}) defining the balls B_{r}(P_{0}) (as in Section 3). We may also estimate φ by integration along ℓ(∂D_{σ_{1}}). (Note that φ(P_{0}) = 0.) We find that ℓ(∂D_{σ_{1}}) ⊂ B_{δ_{1}}(P_{0}) where δ_{1} = c_{1}δ for a fixed constant c_{1}. Since A(1) = 1, we may use the monotonicity formula to show that A(B_{ρ_{0}}(P_{0})) ≤ c_{2}ρ_{0}^{2}, and thus there is a fixed constant ρ_{0} > 0 depending only on c_{1} such that sup_{D_{1}} ρ ∘ ℓ ≥ 2ρ_{0}. By the global Hölder estimate of Proposition 3.4 the map log(c)^{−1/2}ℓ satisfies a uniform Hölder estimate, and thus ℓ satisfies the bound |ℓ(t_{1}) − ℓ(t_{2})| ≤ log(c)^{1/2}c_{3}|t_{1} − t_{2}|^{α} for fixed positive constants c_{3}, α. Using this estimate, we may find a point t_{1} ∈ D_{1} such that ρ(ℓ(t)) ≥ ρ_{0} for all t ∈ D_{1} ∩ D_{δ_{2}}(t_{1}) where δ_{2} = c_{4} log(c)^{−1/(2α)}. We then consider the following function ζ

ζ(P) = \frac{log(ρ(P)/δ_{1})}{log(ρ_{0}/δ_{1})}

for δ_{1} ≤ ρ(P) ≤ ρ_{0}, and ζ(P) = 0 for ρ(P) < δ_{1}, ζ(P) = 1 for ρ(P) > ρ_{0}. Using monotonicity and the coarea formula we can derive the bound ∫_{D_{1}} |∇ζ ∘ ℓ|^{2} dt ≤ c_{5}/ log(1/δ_{1}). On the other hand, if we let r, θ be polar coordinates centered at t_{1}, we may use the fact that ζ ∘ ℓ(t) = 0 for t ∈ D_{σ_{1}} and ζ ∘ ℓ(t) = 1 for t ∈ D_{1} ∩ D_{δ_{2}}(t_{1}) to conclude that ∫_{D_{1}} |∇ζ ∘ℓ|^{2} dt ≥ c_{6}/ log(1/δ_{2}). (This may be seen by using the Schwarz inequality and the fact that ζ∘ℓ changes from 0 to 1 along a fixed fraction of the rays from t_{1} as follows:

1 ≤ (∫_{δ_{2}}^{1} |∇ζ ∘ ℓ| dr)^{2} ≤ (∫_{δ_{2}}^{1} r^{−1} dr) (∫_{δ_{2}}^{1} |∇ζ ∘ ℓ|^{2}r dr) .

The bound then follows by integrating along rays.) Combining these inequalities we conclude that A(1/4) ≥ c_{7}(log(c))^{−β} for constants c_{7}, β depending only on c_{1}. We then observe that if c is chosen larger than a fixed constant, we have c_{7}(log(c))^{−β} ≥ c^{−1}. This completes the proof.

q.e.d.

The next result deals with surfaces which lie near a plane, and shows that such surfaces can be well approximated by smooth conformal maps to the plane. We use the notation Â(ℓ(Σ)∩B_{r}(P)) to denote the monotone quantity ∫_{Σ} F_{r} da from Section 3.

Proposition 4.3. Assume that ℓ = (x(t), y(t), φ(t)) : D_{2} → ℝ^{5} with ℓ(0) = 0 is a stationary legendrian map with A(ℓ(D_{2})) = 1, A(ℓ(D_{1})) ≥ c_{1}^{−1}, and ∫_{D_{2}} |∇y|^{2} dt ≤ ε. Assume further that Â(ℓ(D_{2}) ∩ B_{r}(P)) ≤ (1 + ε)πr^{2} for any P ∈ ℝ^{5}. Given any δ > 0, there is a constant ε_{2} > 0 depending only on δ, c_{1} such that if ε ≤ ε_{2}, then there is a 1-1 holomorphic or anti-holomorphic map x_{0} : D_{1} → ℝ^{2} such that ∫_{D_{1}}(|∇(ℓ − x_{0})|^{2} + |ℓ − x_{0}|^{2}) dt ≤ δ.

Proof. To prove this result by contradiction, it suffices to show that if ℓ^{(j)} is a sequence of maps satisfying the hypotheses with ε_{j} → 0, then a subsequence of ℓ^{(j)} converges strongly in W^{1,2}(D_{1}) to an injective conformal map to the x-plane. By the uniform continuity estimate, we may choose a subsequence, again denoted ℓ^{(j)} which converges uniformly to a continuous W^{1,2} map x_{0} on D_{3/2}. We first show that x_{0} is 1-1 in D_{1}. If this were not true, then we can find a point ξ and points t_{1}, t_{2} ∈ D_{1} with t_{1} ≠ t_{2} and x_{0}(t_{1}) = x_{0}(t_{2}) = ξ. We can show that there is a δ > 0 such that for j sufficiently large (ℓ^{(j)})^{−1}(B_{δ}(ξ)) has distinct connected components U_{1}, U_{2} containing t_{1}, t_{2} respectively. This follows by the same type of capacity argument using the function ζ as in the previous proof, the point being that each connected component of (ℓ^{(j)})^{−1}(B_{δ}(ξ)) must have arbitrarily small diameter if δ is chosen sufficiently small. Since ℓ^{(j)}(t_{i}) → ξ, it follows from monotonicity that each component U_{i}, i = 1, 2 has area at least πδ^{2} − o(j). For j sufficiently large, this contradicts the assumption that Â(ℓ(D_{2})∩B_{δ}(ξ)) ≤ (1+ε_{j})πδ^{2}. Therefore we have shown that x_{0} is 1-1 on D_{1}. It follows that Ω = x_{0}(D_{1}) is an open set bounded by the Jordan curve x_{0}(∂D_{1}).

We now show that ℓ^{(j)} converges strongly to x_{0} in W^{1,2}(D_{1}). We first show that A(ℓ^{(j)}(D_{1})) → A(x_{0}(D_{1})) = A(Ω). To see this, first note that if B_{r}(ξ) is any open ball centered at a point ξ ∈ Ω with ∂Ω ∩ B_{r}(ξ) = ϕ, then we have A(ℓ^{(j)}(D_{1}) ∩ B_{r}(ξ)) → A(Ω ∩ B_{r}(ξ)) = πr^{2}. By the Vitali covering lemma, we can cover almost all of Ω with a disjoint collection of such balls. If O denotes the union of this collection of balls, we then have A(ℓ^{(j)}(D_{1}) ∩ O) → A(Ω ∩ O) = A(Ω). Since Ω \ O may be covered by a finite number of balls B_{r_{k}}(ξ_{k}) with ∑ r_{k}^{2} arbitrarily small, it follows that ℓ^{(j)}(D_{1}) \ O will be contained in this finite union for j sufficiently large, and therefore its area will be arbitrarily small. Thus we have shown that A(ℓ^{(j)}(D_{1})) → A(x_{0}(D_{1})) as required. By lowersemicontinuity of the energy we have the energy of x_{0} on D_{1} less than or equal to the liminf of the energies of the ℓ^{(j)}. Since the ℓ^{(j)} are weakly conformal their energies agree with thier areas. Thus we have the energy of x_{0} on D_{1} is less than or equal to its area. This implies that the two are equal and that x_{0} is weakly conformal. It also implies that there is no energy loss in the convergence of ℓ^{(j)} to x_{0}, and therefore this convergence is strong in W^{1,2}(D_{1}). q.e.d.

We have the following important consequence of these propositions.

Proposition 4.4. Assume that A(ℓ(D_{2})) ≤ c_{1}, and that ℓ_{0} is a linear holomorphic map to a lagrangian plane. Assume further that

∫_{D_{2}}{|∇(ℓ − ℓ_{0})|^{2} + |ℓ − ℓ_{0}|^{2}} dt ≤ δ.

Given any ε > 0, there is a δ > 0 depending only on ε and c_{1}, and a fixed constant c such that if the above inequality holds, then

∫_{D_{1}} |∇(ℓ − ℓ_{0})|^{2} dt ≤ c ∫_{D_{2}} |ℓ − ℓ_{0}|^{2} dt + ε ∫_{D_{2}} |∇(ℓ − ℓ_{0})|^{2} dt.

Proof. We may assume by change of coordinates and dilation that in Darboux coordinates (x, y) the map ℓ_{0}(t) = (t, 0), and we write ℓ(t) = (x(t), y(t)). We begin by estimating the Dirichlet integral of y(t). We observe that the hamiltonian x · y − φ yields the vector field y \frac{∂}{∂y}. We let ζ(x) be a cutoff function with ζ(x) = 1 for |x| ≤ 3/2, and ζ(x) = 0 for |x| ≥ 7/4. We then take the hamiltonian h = ζ(x)(x · y − φ). In proving the desired inequality, we may assume that ∫_{D_{2}} |ℓ − ℓ_{0}|^{2} dt is small, for otherwise the inequality is trivial. Thus it follows that ℓ is uniformly close to ℓ_{0} interior to D_{2}. Thus the hamiltonian h ∘ ℓ has compact support in D_{2}. The vector field determined by h is ζ(x)y \frac{∂}{∂y} + (x · y − φ)ζ_{x} \frac{∂}{∂y}. Applying the first variation formula we get

∫_{D_{2}}⟨∇y, ∇(ζ(x(t))y(t) + (x(t) · y(t) − φ)ζ_{x}(x(t)))⟩ dt = 0

where ∇ refers to derivatives with respect to t. This equation easily implies

∫_{D_{2}} ζ(x(t))|∇y|^{2} dt ≤ c ∫_{D_{2}}(|ζ_{x}| + |ζ_{xx}|)(|y| + |φ|)|∇y||∇x| dt

where we have used the fact that ∇(x · y − φ) = 2y∇x. Replacing ζ with ζ^{4}, this implies

∫_{D_{2}}ζ(x(t))^{4}|∇y|^{2} dt ≤ c ∫_{D_{2}}ζ^{2}(ζ|ζ_{x}|+ζ|ζ_{xx}|+|ζ_{x}|^{2})(|y|+|φ|)|∇y||∇x| dt.

We may then use the arithmetic-geometric mean inequality and a simple choice of ζ (with 0 ≤ ζ ≤ 1) to obtain

∫_{D_{5/4}} |∇y|^{2} dt ≤ c ∫_{D_{15/8}}(|ℓ − ℓ_{0}|^{2} + φ^{2})|∇ℓ|^{2} dt.

Since |∇ℓ|^{2} ≤ 2|∇ℓ_{0}|^{2} + 2|∇(ℓ − ℓ_{0})|^{2}, and |∇ℓ_{0}|^{2} = 1, we have

∫_{D_{5/4}} |∇y|^{2} dt ≤ c ∫_{D_{15/8}}(|ℓ − ℓ_{0}|^{2} + φ^{2}) dt + ε ∫_{D_{2}} |∇(ℓ − ℓ_{0})|^{2} dt (4.3)

where we have used the fact that |ℓ−ℓ_{0}|^{2} +φ^{2} is pointwise small (which follows from Proposition 3.3).

To complete our proof, we now consider the conformality relation for the map ℓ. If we introduce complex coordinates τ = t_{1} + \sqrt{−1}t_{2}, ξ = x_{1} + \sqrt{−1}x_{2}, η = y_{1} + \sqrt{−1}y_{2} then we may write this relation ξ_{τ} ξ_{τ} + η_{τ} η_{τ} = 0. In particular we have

∫_{D_{9/8}} |ξ_{τ} ||ξ_{τ} | dt ≤ ∫_{D_{9/8}} |∇y|^{2} dt. (4.4)

We now define a measurable subset B ⊂ D_{9/8}, the ‘bad set’ to be the set of points for which |ξ_{τ} | ≤ |ξ_{τ} |. Observe that

∫_{D_{9/8}\B} |ξ_{τ} |^{2} dt ≤ ∫_{D_{9/8}} |∇y|^{2} dt. (4.5)

We now estimate the area of ℓ(B). We claim that if t ∈ B, there is a closed disk D^{t} ⊂ D_{5/4} centered at t such that A(D^{t}) ≤ c ∫_{D^{t}} |∇y|^{2} dt. This follows from Proposition 4.3, since for any disk, the reverse inequality implies that ℓ is close to a holomorphic or anti-holomorphic map. Since for small disks centered at t, the map ℓ cannot be close to a holomorphic map, while for the disk of radius 1/8, the map is close to the holomorphic map ℓ_{0}, there must be a disk in which our inequality holds. We consider the collection B of such closed disks, and apply the Besicovitch covering lemma to obtain disjoint subcollections B_{1}, ⋯, B_{q} whose union covers B. Summing the inequalities over each B_{k}, and then summing over k we obtain

∫_{B} |∇ℓ|^{2} dt ≤ cq ∫_{D_{5/4}} |∇y|^{2} dt. (4.6)

Combining the inequalities (4.5) and (4.6) we get

∫_{D_{9/8}} |ξ_{τ} |^{2} dt ≤ c ∫_{D_{5/4}} |∇y|^{2} dt. (4.7)

The standard L^{2} estimate for the Cauchy-Riemann operator then implies

∫_{D_{1}} |∇(x − t)|^{2} dt ≤ c ∫_{D_{9/8}} |ξ_{τ} |^{2} dt + c ∫_{D_{2}} |x − t|^{2} dt.

Combining this inequality with (4.3) and (4.7), we obtain

∫_{D_{1}} |∇(ℓ − ℓ_{0})|^{2} dt ≤ c ∫_{D_{2}}(|ℓ − ℓ_{0}|^{2} + φ^{2}) dt + ε ∫_{D_{2}} |∇(ℓ − ℓ_{0})|^{2} dt. (4.8)

To complete the proof we estimate the L^{2} norm of φ as follows. Since we are free to subtract a constant from φ, we may assume that

∫_{D_{2}}(φ − y · t) dt = 0,

and apply the Poincaré inequality to obtain

∫_{D_{2}}(φ − y · t)^{2} dt ≤ c ∫_{D_{2}} |∇(φ − y · t)|^{2} dt.

Since dφ = xdy−ydx, we see that d(φ−y·t) = yd(x−t)−(x−t)dy−2ydt, and this implies

∫_{D_{2}}(φ − y · t)^{2} dt ≤ c [∫_{D_{2}} y^{2} dt + ∫_{D_{2}} |ℓ − ℓ_{0}|^{2}|∇(ℓ − ℓ_{0})|^{2} dt] ,

and therefore

∫_{D_{2}} φ^{2} dt ≤ c [∫_{D_{2}} y^{2} dt + ∫_{D_{2}} |ℓ − ℓ_{0}|^{2}|∇(ℓ − ℓ_{0})|^{2} dt] .

Combining this inequality with (4.8) then gives the result (using that the pointwise norm of ℓ − ℓ_{0} is as small as we wish). q.e.d.

We now begin the proof of Theorem 4.1. For a map ℓ, a linear conformal map ℓ_{0}, and a radius r, define the following excess-type quantity

E(ℓ, ℓ_{0}, r) = max {r^{−2} ∫_{D_{r}} |∇(ℓ − ℓ_{0})|^{2} dt, r sup_{D_{r}} |∂g| ∘ ℓ}

where ∂g refers to ambient derivatives of g. We will prove the desired result by showing that E decays like a power of r provided that we change ℓ_{0} appropriately as a function of r. We will exploit our freedom to rescale the problem and change coordinates. We may do a unitary change of coordinates so that the lagrangian plane which is the image of ℓ_{0} becomes the x plane. We then write our map as ℓ(t) = (x(t), y(t)). We may also rescale the map so that we may assume that r = 1. We will prove the theorem by establishing the following decay result for E. Observe that the hypotheses of the theorem imply

E(ℓ, ℓ_{0}, r) < ε_{1} (4.9)

for the original map.

Proposition 4.5. There exists ε_{1} > 0 and a number θ ∈ (0, 1/2) such that if (4.9) holds for ε_{1}, then there is a linear conformal map ℓ_{1} with image a lagrangian plane such that

E(ℓ, ℓ_{1}, θ) ≤ \frac{1}{2}E(ℓ, ℓ_{0}, 1). (4.10)

Proof. The proof is by a blow-up argument. We will take θ to be a small constant to be determined. Assume for the sake of contradiction that the conclusion of the theorem fails. We consider sequences ℓ^{(j)}, g^{(j)} with g^{(j)} converging in C^{1} norm to the euclidean metric δ, and

lim_{j→∞} E(ℓ^{(j)}, (t, 0), 1) = 0.

We may write our maps ℓ^{(j)} = (x^{(j)}, y^{(j)}). We then consider the sequence of vector valued maps u^{(j)} = ε_{j}^{−1/2}(x^{(j)} − t) − α_{j} and v^{(j)} =

ε_{j}^{−1/2}y^{(j)} − β_{j} where ε_{j} = E(ℓ^{(j)}, (t, 0), 1). Therefore we have

∫_{D_{1}}(|∇u^{(j)}|^{2} + |∇v^{(j)}|^{2}) dt ≤ 1,

and assume that α_{j}, β_{j} are chosen so that ∫ u^{(j)} = ∫ v^{(j)} = 0. The sequences u^{(j)}, v^{(j)} are bounded in W^{1,2}, so we may choose a subsequence of u^{(j)}, v^{(j)} which converges in L^{2} and weakly in W^{1,2} on the unit disk to limit maps u, v. We now show that u is holomorphic and v = ∇w for a biharmonic function w.

We first consider u. We may use the conformality relation for the map ℓ^{(j)} in the following way. Observe that for the Euclidean metric δ we may write conformality in the complex form \frac{∂x}{∂t} \frac{∂\overline{x}}{∂t} + \frac{∂y}{∂t} \frac{∂\overline{y}}{∂t} = 0 where we are using complex notation x = x^{1} + ix^{2}, y = y^{1} + iy^{2}, t = t^{1} + it^{2}. The conformality relation for ℓ^{(j)} using the metric g^{(j)} is then:

\frac{∂x^{(j)}}{∂t} \frac{∂x^{(j)}}{∂t} + \frac{∂y^{(j)}}{∂t} \frac{∂y^{(j)}}{∂t} = 0.

On the other hand we have:

|∫_{D_{1}} \frac{∂(x^{(j)} − t)}{∂t} \frac{∂(x^{(j)} − t)}{∂t} + \frac{∂y^{(j)}}{∂t} \frac{∂y^{(j)}}{∂t} | ≤ cε_{j}.

Since x = t is holomorphic, we have

\frac{∂x^{(j)}}{∂t} \frac{∂x^{(j)}}{∂t} = \frac{∂(x^{(j)} − t)}{∂t} + \frac{∂(x^{(j)} − t)}{∂t} \frac{∂(x^{(j)} − t)}{∂t}.

We may combine these to obtain

∫_{D_{1}} |\frac{∂(x^{(j)} − t)}{∂t}| dt ≤ cε_{j}.

Thus we may divide by \sqrt{ε_{j}} to obtain

∫_{D_{1}} |\frac{∂u^{(j)}}{∂t} | dt ≤ c\sqrt{ε_{j}}.

It follows that the weak limit u is a (weakly) holomorphic function, as claimed.

To analyze the limit of v^{(j)}, we consider hamiltonians on N of the form h(x), so that

X_{h} = ∑ h_{x_{i}} \frac{∂}{∂y_{i}} .

We use the first variation formula together with the fact that the metric is close to the euclidean metric to obtain

|∫_{D_{1}} ∇y^{(j)} · ∇X^{(j)} dt| ≤ cε_{j} (4.11)

where X^{(j)} = X_{h} ∘ x^{(j)} (and ∇ represents the t derivative). By the Schwarz inequality, and the fact that h is a smooth function of x we have

|∫_{D_{1}} ∇y^{(j)} · DDh∇(x^{(j)} − t)dt| ≤ cε_{j}, (4.12) where D represents derivatives with respect to x. By the chain rule ∇X^{(j)} = DDh · ∇x^{(j)}, so we may combine with the previous inequality to obtain

|∫_{D_{1}} ∇y^{(j)} · DDh dt| ≤ cε_{j}. (4.13)

We define the function h(t) on D_{1} by substituting (t_{1}, t_{2}) for (x_{1}, x_{2}), and observe that the functions DDh then converge to ∇∇h in L^{2} norm. We may then divide by \sqrt{ε_{j}}, and conclude that the weak limit v satisfies

∫_{D_{1}} ∇v · ∇∇h dt = 0 (4.14)

for any C^{2} function h with compact support on D_{1}. On the other hand, the map (x^{(j)}(t), v^{(j)}(t)) is weakly lagrangian for each j. Since this sequence is bounded and converges weakly in W^{1,2}, it follows that the limit map (t, v(t)) is also weakly lagrangian. This condition implies that there is a function w ∈ W^{2,2}(D_{1}) such that v = ∇w. The equation above then shows that w is a biharmonic function.

We may now complete the proof of the proposition by using standard estimates for holomorphic and biharmonic functions. In particular we have for r < 1,

sup_{D_{r}} |∇^{s}u|^{2} ≤ c ∫_{D_{1}} |∇u|^{2} dt (4.15)

for s ≥ 0. Let u_{0} denote the first order part of the Taylor expansion of u at t = 0. Then using the second order Taylor series expansion of u and the fact that u is bounded in L^{2} we have

sup_{D_{r}} |u − u_{0}|^{2} ≤ cr^{4} ∫_{D_{1}} |∇u|^{2} dt. (4.16)

Similarly, let v_{0} denote the first order part of the Taylor expansion of v at 0. Then

sup_{D_{r}} |v − v_{0}|^{2} ≤ cr^{4} ∫_{D_{1}} |∇v|^{2} dt. (4.17)

It follows that

∫_{Dθ}[(u − u_{0})^{2} + (v − v_{0})^{2}] dt ≤ cθ^{6} ∫_{D_{1}}(|∇u|^{2} + |∇v|^{2}) dt ≤ cθ^{6}, and hence for j sufficiently large

∫_{Dθ}[(u^{(j)} − u_{0})^{2} + (v^{(j)} − v_{0})^{2}] dt ≤ cθ^{6}.

Multiplying through by ε_{j}^{2} = E(ℓ^{(j)}, ℓ_{0}^{(j)}, 1) we get

∫_{Dθ}(ℓ^{(j)} − ℓ̂_{1}^{(j)})^{2} dt ≤ cθ^{6}E(ℓ^{(j)}, ℓ_{0}^{(j)}, 1)

where ℓ̂_{1}^{(j)} = ℓ_{0}^{(j)}+ε_{j}(u_{0}+α_{j}, v_{0}+β_{j}). We observe that we can modify ℓ̂_{1}^{(j)} by a term of order 0(ε_{j}^{2}) to make it a conformal map into a lagrangian plane. First note that v_{0} +β_{j} is the gradient of a quadratic function of t, so we can replace v_{0} +β_{j} by the gradient of the same quadratic function of the variable x_{0}^{(j)} + ε_{j}(u_{0} + α_{j}). The resulting perturbation of ℓ̂_{1}^{(j)} is of order ε_{j}^{2}, and makes the image a lagrangian plane. Since x_{0}^{(j)}+ε_{j}(u_{0}+α_{j}) is a holomorphic map, it follows that the Hopf differential of ℓ̂_{1}^{(j)} is of order ε_{j}^{2}. A perturbation of order ε_{j}^{2} will then produce a linear conformal map, which is then the desired map ℓ_{1}^{(j)}. We then have

∫_{Dθ}(ℓ^{(j)} − ℓ_{1}^{(j)})^{2} dt ≤ cθ^{4}(θ^{2} + ε_{j}^{2})E(ℓ^{(j)}, ℓ_{0}^{(j)}, 1) (4.18)

and

∫_{Dθ} |∇(ℓ^{(j)} − ℓ_{1}^{(j)})|^{2} ≤ ∫_{Dθ} |∇(ℓ^{(j)} − ℓ_{0}^{(j)})|^{2} + ∫_{Dθ} |∇(ℓ_{0}^{(j)} − ℓ_{1}^{(j)})|^{2} ≤ cε_{j}^{2}. (4.19)

We may now apply (4.18), (4.19) and Proposition 4.4 to assert that for j sufficiently large

∫_{D_{θ}} |∇(ℓ^{(j)} − ℓ_{1}^{(j)})|^{2} ≤ cθ^{3}E(ℓ^{(j)}, ℓ_{0}^{(j)}, 1)

where θ = \frac{1}{2}θ. On the other hand we clearly have

θ sup_{D_{θ}} |∂g| ∘ ℓ^{(j)} ≤ θE(ℓ^{(j)}, ℓ_{0}^{(j)}, 1).

Thus, for j sufficiently large, the inequality (4.10) holds in any sequence chosen as above. It follows that for ε_{1} sufficiently small the inequality (4.10) holds for a fixed θ. This completes the proof of the proposition.

q.e.d.

Now to complete the proof of the theorem, one iterates the inequality (4.10) with varying center point to show that for any r small and t ∈ D_{1/2}, there is a conformal map ℓ_{t,r} with image a lagrangian plane such that

r^{−2} ∫_{D_{r}(t)} |∇(ℓ − ℓ_{t,r})|^{2} dt ≤ cr^{2α}

for a positive exponent α. This implies that ℓ is C^{1,α} in D_{1/2}. The smoothness now follows from the following result.

Proposition 4.6. Assume that ℓ is a C^{1,α} lagrangian stationary map defined on D = D_{1} with |∇ℓ|(t) ≠ 0 for all t ∈ D. It follows that ℓ is a smooth immersion on D.

Proof. It clearly suffices to show that ℓ is smooth in a neighborhood of t = 0. To do this we assume by choice of Darboux coordinates (x, y) that ℓ(t) = (x(t), y(t)) with x(0) = y(0) = 0 and dy(0) = 0, dx(0) = I. We may then describe a neighborhood of the origin as a C^{1,α} graph y = y(x) with y^{i} = \frac{∂u}{∂x^{i}} for i = 1, 2 for a C^{2,α} function u(x). The stationary condition applied to the hamiltonian vector field X = η_{x} \frac{∂}{∂y} implies

∫ ⟨∇ ∑_{i} y^{i} \frac{∂}{∂y^{i}} , ∇ ∑_{j} η_{j} \frac{∂}{∂y^{j}} ⟩ dt = 0

for any smooth η(x) with compact support. We may do a change of variable to rewrite the equation in the form

∫ ∑_{i,j,k,l} a^{ijkl}(x, u_{x}, u_{xx})u_{x^{i}x^{k}}η_{x^{j}x^{l}} dx = 0

where a^{ijkl}(x, p, q) is a smooth function of its arguments, and a^{ijkl}(0, 0, 0) = δ_{ik}δ_{jl}. Since u is C^{2,α} this is a linear elliptic equation (in divergence form) with C^{0,α} coefficients. In order to get a gain in regularity, we can form the difference quotient u^{(h)}(x) = h^{−1}(u(x + he) − u(x)) where e is any (euclidean) unit vector and h ≠ 0. We then see that u^{(h)} satisfies the equation

∫ ∑(a^{ijkl}(x, u_{x}, u_{xx})u_{x^{i}x^{k}})^{(h)}η_{x^{k}x^{l}} dx = 0.

By a standard manipulation using the Fundamental Theorem of Calculus we have

(a^{ijkl}(x, u_{x}, u_{xx})u_{x^{i}x^{k}})^{(h)}
= a^{ijkl}(x + he, u_{x}(x + he), u_{xx}(x + he))u_{x^{i}x^{k}}^{(h)}
+ (a^{ijkl}(x, u_{x}, u_{xx}))^{(h)}u_{x^{i}x^{k}}
= a^{ijkl}(x)u_{x^{i}x^{k}}^{(h)} + b̃^{jlk}u_{x^{k}}^{(h)} + c^{jl}

where ã, b̃, c̃ are C^{0,α} and a^{ijkl}(0) = δ_{ik}δ_{jl}. We may then apply the divergence form Schauder theory (see [Si2]) for a recent treatment) to get a uniform C^{2,α} estimate on u^{(h)}. This implies that u is a C^{3,α} function in a neighborhood of 0. This argument can be repeated to show that u is smooth. q.e.d.

We now consider the global regularity of minimizing lagrangian maps. We expect the results to hold for stationary maps, but the proof is somewhat more complicated, so for the remainder of this section we assume that our maps are minimizing. The following strong compactness theorem will be important for our methods.

Proposition 4.7. Assume that ℓ_{j} : D → N is a weakly conformal, minimizing lagrangian map for each j = 1, 2, … with A(ℓ_{j}(D) ≤ c for a constant c. There is a subsequence of {ℓ_{j}} which converges strongly in W_{loc}^{1,2}(D) to a minimizing lagrangian map ℓ.

The proof of this result will require the following comparison result.

Lemma 4.8. Suppose that ℓ_{0}, ℓ_{1} : S^{1} → ℝ^{5} are continuous horizontal maps with lengths L_{i} = L(ℓ_{i}(S^{1})) for i = 1, 2, and with sup_{S^{1}} d(ℓ_{0}, ℓ_{1}) ≤ ε ≪ L_{0} + L_{1}. There exists a legendrian map ℓ : S^{1} × [0, 1] :→ ℝ^{5} with ℓ(θ, 0) = ℓ_{0}(θ), ℓ(θ, 1) = ℓ_{1}(θ), and A(ℓ(S^{1} × [0, 1]) ≤ c(L_{0} + L_{1})ε for a constant c.

Proof. Choose an integer N so that (L_{0}+L_{1})/N ∈ [ε, 2ε), and divide S^{1} into N intervals I_{p} = [θ_{p−1}, θ_{p}), p = 1, …, N such that L(ℓ_{0}(I_{p})) + L(ℓ_{1}(I_{p})) = (L_{0} + L_{1})/N. For each p, join the points ℓ_{0}(θ_{p}), ℓ_{1}(θ_{p}) by a horizontal curve ℓ_{p}(t), t ∈ [0, 1] of length bounded by ε. By the isoperimetric inequality Proposition 2.7 we may span the closed curve Γ_{p} = ℓ_{0}(I_{p})∪ℓ_{p} ∪ℓ_{1}(I_{p})∪ℓ_{p−1} with a legendrian disk with area bounded by cε^{2} since the length of this curve is bounded by 4ε. We parametrize this disk on I_{p} × [0, 1] so that it agrees with the specified boundary parametrization. This defines the map ℓ on S^{1} ×[0, 1] with area at most N · cε^{2} ≤ c(L_{0} + L_{1})ε. q.e.d.

We now give the proof of Proposition 4.7.

Proof. We may choose a subsequence also denoted {ℓ_{j}} which converges to a limit ℓ uniformly on compact subsets of D, and such that for almost all r ∈ (0, 1), the sequence L(ℓ_{j}(∂D_{r})) is bounded. We let r_{0} < 1, and we show that ℓ_{j} converges strongly to ℓ in W^{1,2}(D_{r_{0}}), and that ℓ is minimizing in D_{r_{0}}. We may choose r ∈ (r_{0}, 1) so that L(ℓ_{j}(∂D_{r})) ≤ c. Given any ε > 0, we may choose j sufficiently large so that sup_{∂D_{r}} d(ℓ_{j}, ℓ) < ε. We may then use Lemma 4.8 to construct a map ℓ̂ which agrees with ℓ(2t) in D_{r/2}, and agrees with ℓ_{j} on ∂D_{r} such that A(ℓ̂(D_{r})) ≤ A(ℓ(D_{r})) + cε. Since ℓ_{j} is conformal and the area is bounded by the energy, it follows that lim sup E_{D_{r}}(ℓ_{j}) ≤ E_{D_{r}}(ℓ), and hence ℓ_{j} converges strongly to ℓ in W^{1,2}(D_{r}).

To show that ℓ is minimizing in D_{r_{0}}, let ℓ̃ be any W^{1,2} map which agrees with ℓ outside D_{r_{0}}. Construct a map ℓ̃_{j} which agrees with ℓ̃inside D_{r_{0}}, and agrees with ℓ_{j} on ∂D_{r} so that A(ℓ̃_{j}(D_{r})) ≤ A(ℓ̃(D_{r})) + o(j). Since ℓ_{j} is area minimizing, it follows that A(ℓ_{j}(D_{r})) ≤ A(ℓ̃_{j}(D_{r})). Thus, letting j → ∞, we have A(ℓ(D_{r})) ≤ A(ℓ̃(D_{r})) as required. q.e.d.

We now use this result to construct parametrized tangent cones which are again area minimizing. Given any point P ∈ Σ, we may choose conformal coordinates t centered at P, and for any sequence ε_{j} → 0 we consider the map ℓ_{j}(t) = δ_{j}^{−1}ℓ(ε_{j}t) where δ_{j} → 0 is chosen so that A(ℓ_{j}(D_{2})) = 1. There is a constant c_{1} (depending on ℓ) such that A(ℓ(D_{1})) ≤ c_{1} and inf_{t∈D_{1/4}} A(ℓ(D_{1/8}(t))) ≥ c_{1}^{−1}A(ℓ(D_{1})), so by Proposition 4.2 , we have A(ℓ_{j}(D_{1})) ≥ c^{−1}. Furthermore, we have A(ℓ_{0}(D_{2a})) ≤ cA(ℓ_{0}(D_{a})) for a ≥ 1, and hence it follows that A(ℓ_{0}(D_{r})) ≤ cr^{p} for some p > 0 and r ≥ 2. Applying Proposition 4.7 we see that a subsequence of ℓ_{j} converges strongly in W_{loc}^{1,2}(ℝ^{2}) to a nonconstant minimizing, weakly conformal, lagrangian map ℓ_{0} : ℝ^{2} → ℝ^{4}. We have the following properties of ℓ_{0}.

Lemma 4.9. The map ℓ_{0} is a proper map from ℝ^{2} into ℝ^{4}, ℓ_{0}^{−1}(0) = 0, and ℓ_{0}(ℝ^{2}) is a cone in ℝ^{4}.

Proof. We assume that ℓ_{0}(0) = 0, and we show that ℓ_{0}^{−1}(0) is a discrete set of points. To see this we recall the construction of the logarithmic function ζ_{δ}(P) in the proof of Proposition 4.2 which vanishes for ρ(P) ≤ δ, and is equal to 1 for ρ(P) ≥ 1. The function ζ_{δ} ∘ ℓ_{0} then vanishes in a neighborhood of ℓ_{0}^{−1}(0), and the Dirichlet integral (in any disk centered at 0) of ζ_{δ} ∘ ℓ_{0} tends to zero as δ → 0. Since ℓ_{0} is not constant, it follows that the closed set ℓ_{0}^{−1}(0) has zero logarithmic capacity, and hence is a totally disconnected zero dimensional set. To show that it is a discrete set, choose any radius r, and assume by perturbing r slightly if necessary that d(ℓ_{0}(∂D_{r}), 0) = δ > 0. Now if σ < δ, we see from monotonicity that there are a bounded number of components of ℓ_{0}^{−1}(B_{σ}) whose image under ℓ_{0} contains 0. Since this number can only increase as σ decreases, it must remain constant for σ ≤ σ_{0} for some σ_{0} > 0. Each of these components must then contain exactly one point of ℓ_{0}^{−1}(0), and it is a discrete set.

In order to show that ℓ_{0}^{−1}(0) = {0}, we go back to the original map ℓ, and we note that again by monotonicity (upper and lower area bounds), for any σ > 0 there is a fixed bound on the number of components of ℓ^{−1}(B_{σ}) whose image under ℓ intersects B_{σ/2}. For a given σ, denote these components by C_{1}^{(σ)}, …, C_{k}^{(σ)} where C_{1}^{(σ)} is the connected component containing 0. Observe that for ρ < σ, any component C_{j}^{(ρ)} is

contained in one of the components C_{l}^{(σ)}. Thus the number of components increases as σ decreases, so there is a σ_{0} > 0 such that the number of components is constant for σ ≤ σ_{0}. Thus there is a neighborhood Ω of 0 ∈ ℝ^{2} such that for any σ ≤ σ_{0}, there is precisely one component of ℓ^{−1}(B_{σ}) ∩ Ω whose image under ℓ intersects B_{σ/2}. We may now show that ℓ_{0}^{−1}(0) = {0}, for if there were another t_{0} ∈ ℝ^{2} with ℓ_{0}(t_{0}) = 0, there would be σ > 0 such that ℓ_{0}^{−1}(B_{σ}) has at least two connected components whose image contains 0. Thus for j sufficiently large there would be at least two connected components of ℓ_{j}^{−1}(B_{σ}) whose image intersects B_{σ/2}. This contradicts the above condition on ℓ at radius δ_{j}σ, since δ_{j}σ is arbitrarily small when j is large. Thus we have shown that ℓ_{0}^{−1}(0) = {0}.

We now observe that the area A(ℓ_{0}(ℝ^{2})) is infinite. To see this, observe that if it were finite, we could choose an arbitrarily large radius R with the length of the curve ℓ_{0}(∂D_{R}) as small as we choose. The isoperimetric inequality together with the minimizing property of ℓ_{0} would then show that ℓ_{0} is a constant map.

We can now show that the map ℓ_{0} is proper. In fact we claim that there is a constant ε_{0} > 0 such that d(0, ℓ(∂D_{R})) ≥ ε_{0}\sqrt{A(ℓ_{0}(D_{R}))}. To prove this, we argue by contradiction, and suppose there is a sequence R_{i} with d(0, ℓ(∂D_{R_{i}})) ≤ \frac{1}{i} \sqrt{A(ℓ_{0}(D_{R_{i}}))}. We renormalize the map setting ℓ^{(i)}(t) = (\sqrt{A(ℓ(D_{R_{i}}))})^{−1}ℓ_{0}(R_{i}t) so that we now have A(ℓ^{(i)}(D_{1})) = 1 and d(0, ℓ^{(i)}(∂D_{1})) ≤ \frac{1}{i}. We then choose a subsequence, again denoted ℓ^{(i)}, converging strongly in W_{loc}^{1,2}(ℝ^{2}) to a minimizing map ℓ̃. As above the map ℓ̃ is nonconstant and satisfies ℓ̃^{−1}(0) = 0. This contradicts the fact that d(0, ℓ̃(∂D_{1})) = lim d(0, ℓ^{(i)}(∂D_{1})) = 0, and completes the proof of properness.

The fact that ℓ_{0}(ℝ^{2}) is a cone follows from Proposition 3.5. q.e.d.

We now state the main global regularity theorem for minimizing lagrangian maps.

Theorem 4.10. Let ℓ : D_{2} → N be an area minimizing, weakly conformal lagrangian map. There is a finite subset S of D_{1} such that ℓ is a smooth immersion on D_{1} \ S. A point t ∈ S is either a branch point of ℓ, or a singularity at which ℓ has a nonflat tangent cone. The map ℓ is smooth across the branch points, and is Lipschitz at the nonflat singularities.

Proof. We first consider the set of points Ω ⊂ Σ at which every (parametrized) tangent cone is flat; i.e., has image a lagrangian plane. We show that Ω is an open set and that there is a discrete set of points B ⊂ Ω such that ℓ is a smooth immersion on Ω\B. Finally we show that ℓ is a smooth map on all of Ω whose differential vanishes at the points of B. First consider any point P ∈ Ω, and choose coordinates t centered at P. Let ℓ_{0} be any tangent cone at P, and observe that since ℓ_{0}(ℝ^{2}) is a lagrangian plane and ℓ_{0} is an energy minimizing lagrangian map, ℓ_{0} is a smooth harmonic map to its image plane. Since ℓ_{0} is also weakly conformal, it follows that ℓ_{0} is either holomorphic or anti-holomorphic. By changing orientation on the image if necessary, we assume that ℓ_{0} is holomorphic. Since ℓ_{0} is proper and ℓ_{0}^{−1}(0) = {0}, we see that we must have ℓ_{0}(τ) = aτ^{n}, where τ = t_{1} + it_{2}, a is a nonzero complex number and n is a positive integer. If, for any tangent cone n = 1, it follows from Theorem 4.1 that ℓ is a smooth immersion in a neighborhood of P. In any case, we claim that there is a disk D_{r} about 0 such that ℓ is a smooth immersion in D_{r} \ {0}. To verify this, we show that for any sequence t^{(j)} → 0 (t^{(j)} ≠ 0), the map ℓ is a smooth immersion in a neighborhood of t^{(j)} for j sufficiently large. If we had a sequence that violated this condition, we could extract a subsequence again denoted t^{(j)} such that ℓ is not a smooth immersion in any neighborhood of t^{(j)} for all j. We set ε_{j} = |t^{(j)}|, and extract a subsequence so that the corresponding rescaled sequence ℓ_{j}(t) = (δ_{j})^{−1}ℓ(ε_{j}t) converges to a tangent cone. Now for the ℓ_{j}, there is a point t^{(j)}/ε_{j} on the unit circle, near which, ℓ_{j} is not a smooth immersion. Since the tangent map is a smooth immersion on the unit circle, this contradicts Theorem 4.1, and shows that ℓ is a smooth immersion in a deleted neighborhood of any P ∈ Ω. This shows that Ω is open and B is discrete.

Now consider a point P ∈ B, and choose coordinates t centered at P as above. We wish to show that ℓ is a smooth map in a neighborhood of P. We first show that ∇ℓ decays near 0. Precisely, we claim that sup_{D_{r}} |∇ℓ| ≤ cr^{α} for any α ∈ (0, 1). To see this, we show that for r small enough we have sup_{∂D_{r/2}} |∇ℓ| ≤ θ sup_{∂D_{r}} |∇ℓ| for any θ > 1/2. The proof of this is by contradiction, supposing that there is a sequence r_{i} → 0 for which the opposite inequality holds, and forming the corresponding rescaling and tangent map construction. The corresponding tangent map is ℓ_{0} = aτ^{n} where n ≥ 2 (since P ∈ B) and |a| is determined by the condition that A(ℓ_{0}(D_{1})) = 1. By direct computation, for this map it is true that

sup_{∂D_{1/2}} |∇ℓ_{0}| = (1/2)^{n−1} sup_{∂D_{1}} |∇ℓ_{0}|.

On the other hand, since ℓ_{0} is the limit of the rescaled sequence,

sup_{∂D_{1/2}} |∇ℓ_{0}| ≥ θ sup_{∂D_{1}} |∇ℓ_{0}|,

and this gives a contradiction for θ > 1/2. Thus it follows that sup_{∂D_{r/2}} |∇ℓ| ≤ θ sup_{∂D_{r}} |∇ℓ|, and by a simple iteration we get sup_{D_{r}} |∇ℓ| ≤ cr^{α} for any α ∈ (0, 1) as required. It follows that the map ℓ is C^{1,α} in a neighborhood of P for any α ∈ (0, 1).

To get the higher regularity of ℓ near P ∈ B, we need to control the mean curvature near P. We claim that

lim_{r→0} ∫_{D_{r}\D_{r/2}} |H|^{2} dμ = 0 (4.20)

where dμ = λdt is the area form with λ = \frac{1}{2}|∇ℓ|^{2}. In fact, this statement follows from a simple blow-up argument using the facts that:

i) ∫ |H|^{2} dμ is scale invariant.

ii) All tangent maps satisfy H = 0.

iii) The convergence to tangent maps is in the C^{2} topology away from the origin.

We now consider the first order elliptic system satisfied by the mean curvature (see the Appendix). We let σ be the 1-form associated with the mean curvature vector, so that σ(v) = g(ℓ_{∗}(v), JH) for a tangent vector v on ℝ^{2} where H = λ^{−1}Δℓ and Δℓ denoted the Laplacian (tension field) of ℓ. The form σ is then smooth away from t = 0, and satisfies the first order elliptic system

dσ = d ∗ ℓ^{∗}τ + ℓ^{∗}(Ric), δσ = 0. (4.21)

Set

γ = σ − ∗(ℓ^{∗}τ).

Then (4.21) becomes

dγ = ℓ^{∗}(Ric), δγ = ∗(ℓ^{∗}(dτ)). (4.22)

In order to gain regularity, we must show that this system is satisfied distributionally across t = 0. Note that ∫ |σ|^{2} dt = ∫ |H|^{2} dμ since the L^{2} norm of 1-forms is conformally invariant. Now we choose a cutoff function ζ_{r}(t) satisfying ζ_{r}(t) = 0 for |t| ≤ r/2, ζ_{r}(t) = 1 for t ≥ r, |∇ζ_{r}| ≤ 2/r, and let ϕ(t) be any smooth function with compact support in D_{1}. We then have, from (4.22), the equations

∫ δ(ζ_{r}ϕdt) · γ dt = ∫ ζ_{r}ϕℓ^{∗}(Ric)
∫ d(ζ_{r}ϕ) · γ dt = ∫ ζ_{r}ϕℓ^{∗}(dτ).

The first equation implies

|∫ ζ_{r}{δ(ϕdt) · γ dt − ϕℓ^{∗}(Ric)}| ≤ 2(sup |ϕ|)r^{−1} ∫_{D_{r}\D_{r/2}} |σ| dt
+ 2(sup |ϕ|)r^{−1} ∫_{D_{r}\D_{r/2}} |ℓ^{∗}τ|δt.

By the Schwarz inequality, the first term on the right hand side is bounded by a constant times ∫_{D_{r}\D_{r/2}} |σ|^{2} dt which converges to zero by (4.20). The second term clearly converges to zero. It follows from this and the dominated convergence theorem that the first equation of (4.22) is satisfied distributionally. A similar argument applies to establish the weak form of the second equation. Since the right hand side of (4.22) is C^{0,α}, it follows from elliptic regularity theory that γ is C^{1,α} and that therefore σ is C^{0,α}. We now observe that the mean curvature may be written in terms of the components of σ = ∑_{j=1}^{2} σ_{j}(t)dt^{j}:

H = λ^{−1}Δℓ = λ^{−1} ∑_{j=1}^{2} σ_{j}(t)J (ℓ_{∗} (\frac{∂}{∂t^{j}} )) ,

which implies

Δℓ = ∑_{j=1}^{2} σ_{j}(t)J (ℓ_{∗} (\frac{∂}{∂t^{j}} )) .

As an equation for ℓ, we see that since σ_{j} is C^{0,α}, we may conclude that ℓ is C^{2,α}, and then we may feed this information back into (4.22) to get γ in C^{2,α}, and then we may continue inductively to get ℓ smooth in a neighborhood of P.

To complete the proof we must show that Σ \ Ω is a discrete set of points. For this purpose, we consider a point P in this set, and observe that there must be tangent map ℓ_{0} at P whose image is not a plane. Observe that if we take t_{0} ∈ ℝ^{2} \ {0} in the domain of ℓ_{0}, and we take a tangent map ℓ_{1} at t_{0}, then the image of ℓ_{1} is the product of a one dimensional cone in ℝ^{2} with a line. The one dimensional cone is a geodesic in ℝ^{2}, and hence is a line. Therefore the image of ℓ_{1} is a plane. By our previous discussion it follows that ℓ_{0} is a smooth immersion except for a discrete set of points in ℝ^{2} \ {0}. From this it follows that ℓ_{0}(ℝ^{2}) \ {0} is a smooth surface since it is a cone, and the intersection with ∂B_{σ} is an immersed curve for typical values of σ. It follows that ℓ_{0}(ℝ^{2}) is the cone over any such immersed curve. Thus C = ℓ_{0}(ℝ^{2}) is a geometric cone, and is conformally equivalent to ℝ^{2} via some map ξ : C → ℝ^{2}. It follows as above that ξ ∘ ℓ_{0}(τ) = aτ^{n}. In particular, ℓ_{0} is a smooth immersion on all of ℝ^{2} \ {0}. We may then prove as above that ℓ is a smooth immersion on D_{r} \ {0} for some r > 0. It follows that Σ \ Ω is a discrete set as claimed. To prove that ℓ is Lipschitz in a neighborhood of P ∈ Σ\Ω, we need to use the fact that for any tangent map ℓ_{0} at P we have |∇ℓ_{0}|(t)| ≤ c|t|^{α} for some α > 0 (and then we may argue as above). This is a consequence of the structure of the tangent cones discussed in Section 7. In particular, it follows from the fact that the length of the intersection of any such cone with the unit sphere has length strictly greater than 2π. q.e.d.

5. Existence

Let α be a homotopy class of maps [Σ, N] (or a homology class in H_{2}(N, ℤ)). We say a W^{1,2} map f : Σ → N represents α if given any sequence of smooth maps {f_{i}} that strongly approximates f in W^{1,2}, for i sufficiently large each smooth map f_{i} represents α. The fact that the f_{i} lie in the same homotopy class for i sufficiently large follows from the smoothing results of [ScU]. We begin with three lemmas on the homology classes and free homotopy classes of N.

Lemma 5.1. There is a constant C_{0} > 0, depending only on N, such that if f : S^{2} → N is a W^{1,2} map with E(f) < C_{0} then [f] is null in π_{2}(N).

Proof. Suppose not. Then for any c > 0 there is a W^{1,2} map f : S^{2} → N with E(f) < c representing a nonzero element of π_{2}(N). By harmonic map theory [SU1] it follows that there is a harmonic map h : S^{2} → N with E(h) < c representing a nonzero element of π_{2}(N). For c sufficiently small the image of such a harmonic map lies in a coordinate neighborhood and the map is therefore trivial. q.e.d.

Lemma 5.2. Let C be a positive constant. There are at most finitely many free homotopy classes in π_{2}(N) that can be represented by W^{1,2} maps f : S^{2} → N with E(f) < C.

Proof. Suppose not. Then there are infinitely many classes α_{λ} ∈ π_{2}(N), λ ∈ Λ each represented by a W^{1,2} map f_{λ} : S^{2} → N with E(f_{λ}) < C. Minimizing energy in each homotopy class we can represent each α_{λ} by a finite family {f_{λ_{i}} : i = 1, …, n_{λ}} of harmonic maps S^{2} → N with [f_{λ_{i}}] ≠ 0 and ∑_{i}^{n_{λ}} E(f_{λ_{i}}) < C. Since E(f_{λ_{i}}) ≥ C_{0}, there must be infinitely many different free homotopy classes in the set {[f_{λ_{i}}] : λ ∈ Λ, i = 1, …, n_{λ}}. Hence we can assume that the classes α_{λ} are each represented by a harmonic map f_{λ} : S^{2} → N with E(f_{λ}) < C. Choosing a subsequence we can suppose that the {f_{λ}} converge up to bubbling. In particular, the images of the {f_{λ}} converge in Hausdorff distance to the image of a set of harmonic maps of S^{2} and connecting curves. Thus there is a λ_{0} such that for λ > λ_{0} the {f_{λ}} represent the same free homotopy class. The contradiction proves the lemma. q.e.d.

Lemma 5.3. Let C be a positive constant. There are at most finitely many homology classes in H_{2}(N; ℤ) that can be represented by cycles Γ with A(Γ) < C.

Proof. Recall that a cycle Γ determines a closed integral current T_{Γ} by:

T_{Γ}(σ) = ∫_{Γ} σ,

for σ a smooth 2-form of compact support in N. If A(Γ) < C then T_{Γ} satisfies the mass bound M(T_{Γ}) < C. The integral homology class determined by T_{Γ} is [Γ]. The lemma now follows from the compactness theorem of Federer-Fleming and is explicitly stated in [FF, 9.6]. q.e.d.

In this paper, it will be important to use the area functional rather than the energy for the purpose of proving strong convergence and regularity results. This is because we don’t know if such results hold for the energy functional on the space of lagrangian maps. The following proposition will allow us to pass back and forth between these functionals. Let Σ be a smooth closed oriented surface, and α a homotopy class of W^{1,2} lagrangian maps from Σ to N. Let A(α) denote the infimum of area taken over the maps in this class; i.e.,

A(α) = inf{A(ℓ) : ℓ ∈ W_{L}^{1,2}(Σ, N) ∩ α}.

On the other hand, if we fix a metric h on Σ with curvature 1, 0, or −1, then we may consider the infimum E(α, h) given by

E(α, h) = inf{E(ℓ, h) : ℓ ∈ W_{L}^{1,2}(Σ, N) ∩ α}.

Of course, we have A(α) ≤ E(α, h) for any metric h. We will refer to the metrics h above as admissible metrics, and we note that each conformal structure on Σ has an admissible metric which is unique for genus(Σ) > 1, unique up to scale for genus(Σ) = 1, and unique up to conformal transformation if genus(Σ) = 0. The following proposition is fairly standard, but we need a slightly more general version than is common in classical minimal surface theory, so we record it here.

Proposition 5.4. Given any ℓ ∈ W_{L}^{1,2}(Σ, N) and any ε > 0, there exists an admissible metric h and a homeomorphism f of Σ homotopic to the identity so that E(ℓ ∘ f, h) ≤ A(ℓ) + ε. In particular, A(α) = inf_{h}{E(α, h)} where the infimum is taken over all admissible metrics on Σ.

Proof. We consider the pullback metric τ = ∑ τ_{ij}dx^{i}dx^{j} where

τ_{ij} = ⟨ℓ_{∗}(∂/∂x^{i}), ℓ_{∗}(∂/∂x^{j})⟩.

The tensor τ defines an L^{1} degenerate Riemannian metric on Σ. In order to uniformize this metric, we need to make it bounded and positive definite. We let h_{1} be any smooth Riemannian metric on Σ, δ > 0, and set τ_{δ} = τ + δh_{1}. Now, if we choose Λ sufficiently large, and we consider the set

E_{Λ} = {P ∈ Σ : T r_{h_{1}}(τ)(P) > Λ},

then we will have E(ℓ, E_{Λ}, h_{1}) + A(E_{Λ}, h_{1}) < ε/4. We then define τ̂ to be τ_{δ} on Σ \ E_{Λ}, and equal to h_{1} on E_{Λ}. The metric τ̂ may then be uniformized by the measurable Riemann Mapping Theorem of Morrey ([M2]). Thus we may find a homeomorphism f homotopic to the identity, and an admissible metric h such that f^{∗}(τ̂) is conformal to h. Thus it follows that

A(Σ, f^{∗}(τ̂)) = A(Σ, τ̂) = 1/2 ∫_{Σ} T r_{h}(f^{∗}(τ̂)) dμ_{h}.

Now we have A(Σ, τ̂) ≤ A(Σ, τ) + cδ + A(E_{Λ}, h_{1}) < A(ℓ) + cδ + ε/4. The energy of ℓ ∘ f with respect to h taken over f^{−1}(E_{Λ}) is equal to the energy of ℓ with respect to h_{1} taken over E_{Λ} (which is less than ε/4) since f is a conformal map from (f^{−1}(E_{Λ}), h) to (E_{Λ}, h_{1}). On the other hand

E(ℓ∘f, Σ\E_{Λ}, h) = 1/2 ∫_{Σ\E_{Λ}} T r_{h}(f^{∗}(τ)) dμ_{h} ≤ 1/2 ∫_{Σ} T r_{h}(f^{∗}(τ̂)) dμ_{h}.

Combining these results we see that E(ℓ ∘ f, h) ≤ A(ℓ) + cδ + ε/2 which gives the desired conclusion for δ small enough. q.e.d.

Minimizing in a homotopy class

Let Σ be a compact surface and let α ∈ [Σ, N] be a lagrangian homotopy class. Let h be an admissible metric on Σ for which E(α, h) = A(α). Note that if Σ = S^{2}, then any constant curvature one metric has this property. Denote the induced volume form by dζ. Let {ℓ_{i}} be an energy minimizing sequence of W^{1,2} lagrangian maps Σ → N representing α. Choosing a subsequence we can suppose that {ℓ_{i}} converges weakly in W^{1,2}(Σ, N), strongly in L^{2}(Σ, N) and pointwise almost everywhere to a weakly lagrangian map ℓ ∈ W^{1,2}(Σ, N). Of course the convergence may not be strongly in W^{1,2}. To keep track of the energy loss we describe the convergence in measure theoretic language. Consider the energy measures, ζ_{i} = e(ℓ_{i})dζ = \frac{1}{2}|∇ℓ_{i}|^{2}dζ, on Σ. Using the weak convergence theorem for measures and choosing a subsequence we can suppose that these measures converge weakly to a limit measure η. Denote the energy measure associated to the limit map ℓ by, η_{ℓ} = e(ℓ)dζ. Set

η = η_{ℓ} + η_{Z}.

η_{Z} is a measure, with support Z, that records the failure of strong convergence. We will call η_{Z} the defect measure. We will show in the next propositions that if ℓ_{i} is an area minimizing sequence, then the defect measure is a finite sum of point masses.

Proposition 5.5. Assume that h is an admissible metric on Σ for which E(α, h) = A(α), and suppose that {ℓ_{i}} is a sequence of weakly lagrangian maps in W^{1,2}(Σ, N) that minimize energy in a lagrangian homotopy class α. Let ℓ be the weak limit. There is an ε_{0} > 0, depending only on N, such that if D_{r} ⊂ Σ is a disc of radius r and E_{D_{r}}(ℓ_{i}) < ε_{0} then a subsequence (still denoted {ℓ_{i}}) converges strongly to ℓ in W^{1,2}(D_{\frac{r}{2}} , N).

Proof. Let η be the limit energy measure. For any ε > 0 there is at most a finite set S of points, such that if Q ∈ S then η({Q}) ≥ ε. For P ∉ S there is an r_{0} > 0 such that if r < r_{0} and i is sufficiently large then E_{D_{r}(P)}(ℓ_{i}) ≤ 2ε. Fix such a radius r and suppose that for any subsequence the convergence on D_{\frac{r}{2}} (P) results in energy loss. In particular there is a δ > 0 such that

E_{D_{\frac{r}{2}}} (ℓ) + δ < lim inf_{i} E_{D_{\frac{r}{2}}} (ℓ_{i}).

We can choose ρ ∈ [r/2, r] and a subsequence of {ℓ_{i}} ( that we continue to denote \{ℓ_{i}\}) such that ℓ_{i}_{|_{∂Dρ}} has energy bounded by \frac{2ε}{r}. It follows that ℓ_{i}_{|_{∂Dρ}} is continuous, rectifible and has length bounded by a uniform constant times ε. Thus a subsequence converges uniformly to ℓ_{|_{∂Dρ}}. Using the collar construction of Lemma 4.8, for each sufficiently large i, we can construct a map ℓ̃_{i} ∈ W^{1,2}(Σ, N) that agrees with ℓ on D_{\frac{r}{2}} , agrees with ℓ_{i} outside D_{r} and with area

A(ℓ̃_{i}) + \frac{δ}{2} < E(ℓ_{i}).

The maps ℓ̃_{i} and ℓ_{i} differ by a W^{1,2} map s_{i} : S^{2} → N with A(s_{i}) < cε, where c is a uniform constant. By Proposition 5.4, this map can be reparametrized to have energy less than 2cε. Choose ε < C_{0}/(2c), where C_{0} is the constant in Lemma 5.1 . Then s_{i} has energy less than C_{0} and hence is trivial in homotopy. Thus {ℓ̃_{i}} is a sequence in the same lagrangian homotopy class as {ℓ_{i}} but with strictly smaller area, contradicting the area minimizing property of {ℓ_{i}}. q.e.d.

The following corollary is an immediate consequence.

Corollary 5.6. Assume that h is an admissible metric on Σ for which E(α, h) = A(α), and suppose that {ℓ_{i}} is a sequence of weakly lagrangian maps in W^{1,2}(Σ, N) that minimize energy in a lagrangian homotopy class α. Suppose ℓ be the weak limit and that the energy measures ζ_{i} = e(ℓ_{i})dζ converge weakly as measures to η. Then

η = η_{ℓ} + η_{Z}.

Z, the support of η_{Z}, consists of a finite number of points {x_{1}, …, x_{k}} ⊂ Σ and

η_{Z} = ∑_{j=1}^{k} m_{x_{j}}δ_{x_{j}},

where the masses m_{x_{j}} ≥ ε_{0}. The constant ε_{0} is given in Proposition 5.5.

Proposition 5.7. Each point x ∈ Z can be used to construct a finite set of lagrangian stationary, weakly lagrangian maps ℓ_{x_{λ}} ∈ W^{1,2}(S^{2}, N). Each map, ℓ_{x_{λ}}, is minimizing in some free lagrangian homotopy class. The construction allows no energy loss:

lim inf_{i} E(ℓ_{i}) = E(ℓ) + ∑_{x_{λ}} E(ℓ_{x_{λ}}).

Proof. Let x ∈ Z. There is a sequence of radii r_{i} → 0 such that

lim inf_{i} E_{D_{r_{i}}(x)}(ℓ_{i}) ≥ ε_{0},

since otherwise, the convergence in a neighborhood of x is strong. By the Courant-Lebesgue Lemma we can suppose that L(ℓ_{i}(∂D_{r_{i}})) < ε_{i} where the ε_{i} → 0. Set C_{r_{i}} = ∂D_{r_{i}}(x). We can suppose that the curves {ℓ_{i}(C_{r_{i}})} all lie in Darboux coordinate neighborhoods U_{i}. The lagrangian isoperimetric inequality, Proposition 2.7 then implies that there are lagrangian maps m_{i} : D → U_{i} of the unit disc D with ∂m_{i}(D) = ℓ_{i}(C_{r_{i}}) and with

A(m_{i}(D)) ≤ C (L(ℓ_{i}(C_{r_{i}})))^{2} .

The constant C is universal. Rescale the maps ℓ_{i}_{|_{Dr_{i}}} so that they are defined on the unit disc. By stereographic projection we consider the rescaled ℓ_{i} defined on the southern hemisphere of S^{2} and the m_{i} defined on the northern hemisphere. Denote these maps (ℓ_{x})_{i} : S^{2} → N. Note that E((ℓ_{x})_{i}) ≥ ε_{0} and so each (ℓ_{x})_{i} is nontrivial. Choosing a subsequence we can suppose that each (ℓ_{x})_{i} represents a fixed lagrangian homotopy class α_{x}. Then, since the original sequence is minimizing and A(m_{i}(D)) → 0,the sequence {(ℓ_{x})_{i}} must be minimizing. To each map (ℓ_{x})_{i} we can associate an energy measure (ζ_{x})_{i} = e((ℓ_{x})_{i})dζ. We next balance each map (ℓ_{x})_{i} as follows: Consider the domain S^{2} = {x⃗ ∈ ℝ^{3} : |x⃗| = 1}. Reparametrize each (ℓ_{x})_{i} so that the center of mass of the measure (ζ_{x})_{i} on S^{2} is the origin of ℝ^{3}. We call such a map and its associated energy measure balanced. We will denote the balanced sequence {(ℓ_{x})_{i}}.

Choosing a subsequence we can suppose that the balanced sequence {(ℓ_{x})_{i}} converges weakly in W^{1,2}(S^{2}, N), strongly in L^{2}(S^{2}, N) and pointwise almost everywhere to a limit weakly lagrangian map ℓ_{x} ∈ W^{1,2}(S^{2}, N). The measures (ζ_{x})_{i} converge to a balanced measure ζ_{x}. If the convergence to ℓ_{x} is strong the lemma is proved. If not, we consider the points where strong convergence fails and iterate the argument. If at each step the map ℓ_{x} is nontrivial then, since the total energy is bounded, the process must terminate. If ℓ_{x} is trivial then the balanced condition implies that there are at least two points in the support of the defect measure of ζ_{x} each with mass ≥ ε_{0} and thus again the process must terminate after finitely many steps. Note that since A(m_{i}(D)) → 0 there is no energy loss. Clearly ℓ and each map ℓ_{x_{λ}} minimizes energy in some free lagrangian homotopy class. q.e.d.

Using the standard nomenclature we will call the maps ℓ_{x_{λ}} : S^{2} → N, bubbles, and the limit process bubbling.

Let Σ denote a Riemann surface and f ∈ W^{1,2}(Σ, N). Then f determines a current T_{f} as follows: Let τ be a smooth 2-form on N with compact support. Define

T_{f}(τ) = ∫_{Σ} f^{∗}(τ).

We claim that T_{f} is an integral current and as such it represents a class [f] ∈ H_{2}(N; ℤ). To verify this, first note that if a sequence {f_{i}} in W^{1,2}(Σ, N) converges strongly to f then the currents T_{f_{i}} converge to T_{f} in the weak (current) topology. Since W^{1,2}(Σ, N) is dense in C^{∞}(Σ, N) in the strong W^{1,2}-topology there is a sequence {f_{i}} of smooth maps Σ → N that converge strongly in W^{1,2} to f. The currents {T_{f_{i}}} are integral currents that satisfy a mass bound. By the compactness theorem [F] they converge in the flat norm topology to an integral current T. It then follows from [Si1, 31.2] that T = T_{f}. Thus, T_{f} is an integral current. Suppose that {f_{i}} is a sequence of maps with ∥f_{i}∥_{W^{1,2}} < E that converges strongly to f ∈ W^{1,2}(Σ, N) and that [f_{i}] = α for all i. Then, as above, the integral currents T_{f_{i}} converge in the flat norm topology to an integral current T_{f}. By a result of Federer-Fleming [FF] the current T_{f} also represents α in H_{2}(N; ℤ). In other words, for maps from surfaces, strong W^{1,2} convergence preserves homology.

Suppose that Σ = S^{2}. Let {ℓ_{i}} be an area minimizing sequence of lagrangian maps S^{2} → N representing a free homotopy class and let ℓ be the weak limit. Note that ℓ is stationary under compactly supported variations of the domain, since such variations preserve the lagrangian condition. It follows by a standard argument (see, for example, [S]) that the Hopf differential is holomorphic and hence, since the domain is S^{2}, that the Hopf differential vanishes. In particular, ℓ is weakly conformal.

Theorem 5.8. Let α ∈ π_{2}(N) be a lagrangian homotopy class. Let {ℓ_{i}} be an area minimizing sequence of W^{1,2} lagrangian maps S^{2} → N each representing α. Then a subsequence of {ℓ_{i}} converges without energy loss to a finite collection {ℓ_{λ}}_{λ∈Λ} of lagrangian stationary, weakly conformal, weakly lagrangian maps in W^{1,2}(S^{2}, N). Each map ℓ_{λ} minimizes area among W^{1,2} lagrangian maps in some free homotopy class and

∑_{λ}[ℓ_{λ}] = α,

where [ℓ_{λ}] denotes the homology class determined by the map ℓ_{λ}.

Proof. The first two statements follow from the proof of Proposition 5.7. To prove the third statement we first consider the case where the limit consists of a single map ℓ ∈ W^{1,2}(S^{2}, N). In particular, a subsequence of {ℓ_{i}} converges strongly in W^{1,2}(S^{2}, N) to ℓ. Thus, as discussed above, the homology class is preserved in the limit. Next consider the case where the limit consists of more than one map ℓ_{λ} ∈ W^{1,2}(S^{2}, N). By construction, each map ℓ_{λ} is the weak W^{1,2} limit of a minimizing sequence (ℓ_{λ})_{i} of lagrangian maps S^{2} → N. Since there is no energy loss, each sequence must converge strongly. Note that, for each i, the homology class ∑_{λ}[(ℓ_{λ})_{i}] is equal to the class α. Thus,

∑_{λ}[ℓ_{λ}] = ∑_{λ}[(ℓ_{λ})_{i}] = α.

q.e.d.

Non-collapsible surfaces

Let Σ be a Riemann surface with conformal structure μ and let ϕ : Σ → N be a smooth lagrangian map. We will say that a map ℓ : Σ → N has the same action on π_{1} as ϕ if for p ∈ Σ there is a path ρ from ℓ(p) to ϕ(p) such that

ℓ_{∗} = ρ_{∗} ∘ ϕ_{∗} ∘ ρ_{∗}^{−1},

where ϕ_{∗} : π_{1}(Σ, p) → π_{1}(N, ϕ(p)) and ℓ_{∗} : π_{1}(Σ, p) → π_{1}(N, ℓ(p)) are the induced maps on π_{1}. We recall that in [SY] it is shown that the induced map on π_{1} is well-defined for any map in W^{1,2}(Σ, N) and that the induced map on π_{1} is preserved in the weak limit. We define

L_{ϕ} = {ℓ ∈ W^{1,2}(Σ, N) : ℓ is weakly lagrangian and

ℓ has the same action on π_{1} as ϕ}.

Set

E_{μ} = inf{E(ℓ, μ) : ℓ ∈ L_{ϕ}}.

Proposition 5.9. There exists an ℓ ∈ L_{ϕ} such that E(ℓ) = E_{μ}.

Proof. Let {ℓ_{i}} ∈ L_{ϕ} be an energy minimizing sequence. Choosing a subsequence we can suppose that {ℓ_{i}} converges weakly in W^{1,2}(Σ, N), strongly in L^{2}(Σ, N) and pointwise almost everywhere to a weakly lagrangian map ℓ ∈ W^{1,2}(Σ, N) with the same induced action on π_{1} as ϕ. Since the class L_{ϕ} is weakly closed, we have ℓ ∈ L_{ϕ}, and hence lim E(ℓ_{i}) ≤ E(ℓ) = E_{μ}. It follows that there is no energy loss, and hence ℓ_{i} converges strongly to ℓ in W^{1,2}. q.e.d.

Suppose that ϕ : Σ → N is a piecewise C^{1} lagrangian map. Let γ : S^{1} → Σ be an oriented simple closed curve on Σ. Suppose that ϕ_{∗}([γ]) = 1, where ϕ_{∗} is the induced map on π_{1}. The following discussion applies to any such curve γ but is only interesting when [γ] is nontrivial in π_{1}(Σ). Also, we could carry through the following discussion for any closed curve on Σ with similar results. We confine ourselves to simple closed curves because they are suited to our applications. We define the period of γ as follows. Let D be an oriented immersed disc in N with ∂D = ϕ(γ). Set:

period_{D}(ϕ, γ) = ∫_{D} ω.

This is well-defined independent of the choice of oriented representative γ ∈ [γ]. If D′ is another oriented immersed disc in N with ∂D′ = ϕ(γ) the pair D, D′ determine an element α of π_{2}(N). Then period_{D}(γ) − period_{D′}(γ) = ω(α). The period of γ is thus well-defined only modulo the values of ω on π_{2}(N). If the values of ω on π_{2}(N) are rationally related then we can define:

period(ϕ, γ) = inf_{{D: ∂D=ϕ(γ)}} |∫_{D} ω| .

However, if the values of ω on π_{2}(N) are not rationally related then this definition yields period(ϕ, γ) = 0. Accordingly, in the general case, we choose a constant C > 0 and consider:

per_{C}(ϕ, γ) = inf_{{D:∂D=ϕ(γ),A(D)<C}} |∫_{D} ω| .

If there are no discs D ⊂ N with ∂D = ϕ(γ) and A(D) < C then we set per_{C}(ϕ, γ) = ∞. Because we restrict the area of the spanning disc to be less than C the integral of ω over a spanning disc is well-defined modulo the values of ω on classes in π_{2}(N) that can be represented by two-spheres with energy less than 2C. By Lemma 5.2 there are at most finitely many such classes. Hence the infimum is taken over finitely many classes. In particular, if per_{C}(ϕ, γ) = 0 then there is an immersed disc D with ∂D = ϕ(γ) and ∫_{D} ω = 0. Unfortunately per_{C}(ϕ, γ) depends on the representative γ ∈ [γ]. Define:

period_{C}(ϕ, [γ]) = inf_{γ∈[γ]} per_{C}(ϕ, γ).

Consider the energy functional E(S^{2}, N) on piecewise C^{1} maps S^{2} → N. We will require that 2C be a regular value of E(S^{2}, N). The next lemma describes the behavior of period_{C} under hamiltonian variations.

Lemma 5.10. Let Σ be a compact surface. Let [γ] be a nontrivial homotopy class that can be represented by a simple closed curve.

Suppose that ϕ : Σ → N is a piecewise C^{1} lagrangian map such that ϕ_{∗}([γ]) = 1. Let h_{t}, 0 ≤ t ≤ 1 be a 1-parameter family of hamiltonian diffeomorphisms with h_{0} = Id. Then there is an ε > 0 such that if t < ε then:

period_{C}(h_{t} ∘ ϕ, [γ]) = period_{C}(ϕ, [γ]).

Proof. Choose a simple closed curve γ_{0} on Σ that represents [γ]. Set

S(ϕ, γ_{0}, C) = {| ∫_{D} ω| : A(D) < C, ∂D = ϕ(γ_{0})}.

For any C this set of nonnegative real numbers is finite. For any simple closed curve γ on Σ that represents [γ] we have:

S(ϕ, γ, C) ⊂ S(ϕ, γ_{0}, C + A(ϕ(Σ))).

It follows that ∪_{γ∈[γ]}S(ϕ, γ, C) is a finite set of values and each value is achieved by a simple closed curve in [γ]. Denote a set of such curves by {γ_{λ} : λ ∈ Λ} and a set of spanning discs by {D_{λ} : λ ∈ Λ}, where Λ labels the elements of ∪_{γ∈[γ]}S(ϕ, γ, C). Note that under hamiltonian isotopy the period of a curve does not change. Thus the set ∪_{γ∈[γ]}S(ϕ, γ, C) can change under h_{t} only because of a change in the area of the spanning discs. Clearly for sufficiently small t and each λ ∈ Λ, A(h_{t}(D_{λ})) < C. It follows that for sufficiently small t,

∪_{γ∈[γ]}S(ϕ, γ, C) ⊂ ∪_{γ∈[γ]}S(h_{t} ∘ ϕ, γ, C).

By the choice of C there is a δ > 0 such that the classes in π_{2}(N) that can be represented by piecewise C^{1} maps with energy less than 2(C + δ) are the same as the classes that can be represented piecewise C^{1} maps with energy less than 2(C − δ). It follows that for each closed curve γ on Σ that represents [γ] we have:

S(ϕ, γ, C) = S(ϕ, γ, C + δ).

Thus for sufficiently small t,

∪_{γ∈[γ]}S(h_{t} ∘ ϕ, γ, C) ⊂ ∪_{γ∈[γ]}S(ϕ, γ, C).

The result follows. q.e.d.

Remark. The lemma remains true for any 1-parameter family of diffeomorphisms that preserves the lagrangian condition and preserves the periods of curves on the lagrangian surface. In particular, the lemma is true for 1-parameter families with variational vector field X of the type used in the proof of the monotonicity formula. Such vector fields have support in a disc on Σ and have lifts that are contact vector fields. Thus they satisfy: d(X ω)_{|_{Σ}} = 0 and therefore X ω is a closed 1form on Σ with support in a disc. It follows that ∫_{γ} X ω = 0 for any closed curve γ. This implies that the period of γ is preserved. On the other hand the lemma is not true for arbitrary 1-parameter families of symplectic diffeomorphisms since these do not preserve the periods of curves.

Corollary 5.11. If period_{C}(ϕ, [γ]) = 0 then there is a simple closed curve γ ∈ [γ] and an immersed disc D with ∂D = ϕ(γ) and ∫_{D} ω = 0.

Proof. From the proof of the lemma we have that ∪_{γ∈[γ]}S(ϕ, γ, C) is a finite set of values and each value is achieved by a simple closed curve in [γ]. The corollary follows. q.e.d.

The period_{C} invariant is defined for weakly lagrangian maps in W^{1,2}(Σ, N). Let ℓ : Σ → N be such a map and suppose ℓ_{∗}([γ]) = 1. Let C_{r}^{0}(S^{1}, N) denote the curves S^{1} → N that are continuous and rectifiable. Define:

period_{C}(ℓ, [γ]) = inf_{{γ∈[γ]: ℓ∘γ∈C_{r}^{0}(S^{1},N)}} per_{C}(ℓ, γ).

This invariant is preserved under weak convergence. In particular, let {ℓ_{i}} be a sequence of weakly lagrangian maps in W^{1,2}(Σ, N) with ℓ_{i}_{∗}([γ]) = 1 for all i and E(ℓ_{i}) < E for some constant E. Suppose that period_{C}(ℓ_{i}, [γ]) = p for each i. Let ℓ be the weak limit in W^{1,2}(Σ, N) of a subsequence of {ℓ_{i}} (that we will continue to denote {ℓ_{i}}). Let γ_{0} ∈ [γ] be a simple closed curve such that ℓ(γ_{0}) is continuous and rectifiable. Let T be a tubular neighborhood of γ_{0} in Σ with Γ : S^{1} × [−1, 1] → T a smooth immersion. For s ∈ [−1, 1], let γ_{s} : S^{1} → Σ be the curve γ_{s}(t) = Γ(t, s). For almost every s, the curve ℓ(γ_{s}) is continuous and rectifiable. Given any ε > 0, if s is sufficiently small, the curves ℓ(γ_{0}) and ℓ(γ_{s}) bound an annulus of area less than ε. Moreover, for almost every s, {ℓ_{i}(γ_{s})} is continuous, rectifiable and has uniformly bounded length. Thus, there is a subsequence of {ℓ_{i}} (that we will continue to denote {ℓ_{i}}) that converges uniformly to ℓ(γ_{s}). In particular, for i sufficiently large, ℓ_{i}(γ_{s}) is uniformly close to ℓ(γ_{s}). Therefore, for i sufficiently large, the curves ℓ_{i}(γ_{s}) and ℓ(γ_{0}) bound an annulus of area less than 2ε. It follows that:

period_{C}(ℓ, [γ]) ≤ p.

The proof of the opposite inequality is similar.

Remarks.

(i) Lemma 5.10 applies to weakly lagrangian maps in W^{1,2}(Σ, N). The proof is essentially the same.

(ii) Suppose that {ℓ_{i}} is a minimizing sequence of maps in W^{1,2}(Σ, N) all with the same period_{C} invariants and with weak limit ℓ. Then it follows that ℓ also has these period_{C} invariants. The replacement argument of Proposition 5.5 can be made to preserve the period_{C} invariants and therefore constructs an admissible comparison map among maps with fixed periods. It follows that a subsequence of {ℓ_{i}} converges strongly to ℓ on Σ \ S, where S is a finite set of points. The details are left to the reader.

Let Σ be a closed surface of genus greater than one. Fix δ > 0. Suppose that ϕ : Σ → N is a C^{1} lagrangian map such that either ϕ_{∗} : π_{1}(Σ, p) → π_{1}(N, ϕ(p)) contains no nontrivial element represented by a simple closed curve in its kernel or if such an element [γ] does exist then period_{C}(ϕ, [γ]) ≥ δ > 0. We require that C satisfy 2A(ϕ) < C and that 2C is a regular value of the energy functional E(S^{2}, N) on piecewise C^{1} maps S^{2} → N. We call such a lagrangian map non-collapsible. The notion is a generalization of incompressibility as formulated in [SU2].

In analogy to the above definitions we define:

L̃_{ϕ} = {ℓ ∈ W^{1,2}(Σ, N) : ℓ is weakly lagrangian,

ℓ has the same action on π_{1} as ϕ and the period_{C} invariants of ℓ and ϕ are equal}.

Set

Ẽ_{μ} = inf{E(ℓ, μ) : ℓ ∈ L̃_{ϕ}}.

Proposition 5.12. There exists an ℓ ∈ L̃_{ϕ} such that E(ℓ, μ) = Ẽ_{μ}.

Proof. Same as above. q.e.d.

Suppose that genus(Σ) = g ≥ 2. Denote the Teichmüller space of Σ by T (Σ). T (Σ) can be described as the equivalence classes of pairs (Σ_{1}, f_{1}) where f_{1} : Σ → Σ_{1} is a homeomorphism and two pairs (Σ_{1}, f_{1}) and (Σ_{2}, f_{2}) are equivalent if f_{2}∘f_{1}^{−1} is homotopic to a conformal map. We can mark Σ by choosing a set of simple closed curves on Σ whose homotopy classes generate π_{1}(Σ). A marked surface represents an element of Teichmüller space under the equivalence relation that two marked surfaces are equivalent if there is a homeomorphism respecting the markings and homotopic to a conformal map. Let R(Σ) denote the Riemann moduli space of conformal structures on Σ. Then R(Σ) = T (Σ)/Mod(Σ) where Mod(Σ), the Teichmüller modular group of Σ, is the mapping class group modulo its subgroup of ineffective elements. The following is a result of Abikoff [A], (quoted in [SU2]).

Theorem 5.13. There exists a compactification R(Σ) of R(Σ) such that the boundary points R(Σ) \ R(Σ) correspond precisely to the Riemann surfaces with nodes that can be obtained from Σ by collapsing a set of admissible closed curves on Σ.

Let Σ_{0} denote a fixed Riemann surface of genus g ≥ 2 and let Σ_{0} be the base surface of T (Σ_{0}). We define an energy function on Teichmüller space as follows: Given a lagrangian map ϕ : Σ_{0} → N consider a point μ = (Σ, f) ∈ T (Σ_{0}). Define:

Ẽ_{μ} = inf{E(ℓ, μ) : ℓ ∘ f ∈ L̃_{ϕ}}.

The conformal invariance of E insures that Ẽ_{μ} is well-defined independent of the representative of μ. This defines a map:

Ẽ : T (Σ_{0}) → ℝ
μ ↦ Ẽ_{μ}.

This function is lower semi-continuous.

Proposition 5.14. Ẽ_{μ} is attained at each μ ∈ T (Σ_{0}) by some weakly lagrangian W^{1,2}(Σ, N) map ℓ_{μ} which has the same induced map on π_{1}(Σ) and the same period_{C} invariants as ϕ.

Theorem 5.15. Let ℓ_{i} : Σ_{μ_{i}} → N be a sequence of maps in W^{1,2} for the conformal structure μ_{i} on Σ such that E(ℓ_{i}, μ_{i}) < B. Suppose that π(Σ_{μ_{i}}) → Σ_{∞} where π : T (Σ_{0}) → R(Σ_{0}) is the quotient map and Σ_{∞} ∈ R(Σ_{0}) \ R(Σ_{0}). Then for i sufficiently large, (ℓ_{i})_{∗}([γ]) = 1 for at least one [γ] ∈ π_{1}(Σ) represented by a simple closed curve. Moreover,

lim inf_{i} period_{B}(ℓ_{i}, [γ]) = 0.

Proof. The argument is based on the proof of a similar result in [SU2]. The sequence Σ_{μ_{i}} gives a deformation of Σ_{0} onto a Riemann surface Σ_{∞} with nodes w_{m}, m = 1, …, r, where each node corresponds to the pinching of a homotopically nontrivial simple closed curve γ_{m}. There is a sequence {D_{j}^{m}} of closed annular neighborhoods of γ_{m} such that D_{j}^{m} converges to the node w_{m} of Σ_{∞} and, for each fixed j, the change in conformal structure on Σ_{0} as Σ_{μ_{i}} → Σ_{∞} is restricted to the interior of ∪_{m=1}^{r}D_{j}^{m}. Let Σ_{j} = Σ_{0} \ ∪_{m}D_{j}^{m}. Then, for fixed j, {ℓ_{i}_{|_{Σ_{j}}} } is a sequence of W^{1,2} maps with bounded μ_{j} energy. Choosing a subsequence of the sequence {ℓ_{i}}, that we continue to denote {ℓ_{i}}, we can suppose that {ℓ_{i}_{|_{Σ_{1}}} } converges weakly in W^{1,2}, strongly in L^{2} and pointwise almost everywhere to a W^{1,2} map ℓ^{(1)} : Σ_{1} → N. Choosing successive subsequences for, j = 2, 3, …, we construct a sequence of W^{1,2} maps ℓ^{(j)} : Σ_{j} → N, with the property that ℓ^{(k)} extends ℓ^{(j)} from Σ_{j} to Σ_{k} for each k > j. Let j → ∞. We get a W^{1,2} map ℓ : Σ′_{∞} → N, where Σ′_{∞} is the punctured Riemann surface Σ_{0} \ {w_{1}, …, w_{r}} and ℓ^{(j)} → ℓ strongly in W_{loc}^{1,2}. Since E(Σ_{j}, μ_{j}) < B it follows that E(ℓ) < B. Trivially we can extend ℓ to a W^{1,2} map Σ̃_{∞} → N where Σ̃_{∞} = Σ′_{∞} ∪ {(x_{1}, y_{1}), …, (x_{r}, y_{r})} is the closed Riemann surface obtained by adding the pair of points (x_{m}, y_{m}) at the two punctures of Σ′_{∞} corresponding to each node w_{m} ∈ Σ_{∞}.

Fix m. Let γ be a curve homotopic to γ_{m} and contained in D_{j}^{m} for some large j. Then γ ⊂ Σ_{∞} is homotopically trivial. Let T be a tubular neighborhood of γ in Σ_{∞} \ {w_{m}} with Γ : S^{1} × [−1, 1] → T a smooth immersion. For s ∈ [−1, 1], let γ_{s} : S^{1} → Σ be the curve γ_{s}(t) = Γ(t, s). For almost every s, ℓ(γ_{s}) is continuous, rectifiable and homotopically trivial. Also, for almost every s, there is a subsequence of {ℓ_{i}}, that we continue to denote {ℓ_{i}}, such that the curves {ℓ_{i}(γ_{s})} converge uniformly to ℓ(γ_{s}). Therefore, for sufficiently large i and almost every s, ℓ_{i}(γ_{s}) is homotopically trivial.

We next show that lim inf_{i} period_{B}(ℓ_{i}, [γ]) = 0. Choose any ε > 0. Then γ ⊂ Σ_{∞} can be chosen so that ℓ(γ) is continuous and rectifiable and so that a subsequence of the curves {ℓ_{i}(γ)} converges uniformly to ℓ(γ). ℓ(γ) bounds the image of a disc D under the weakly lagrangian map ℓ. Clearly, E(ℓ_{|_{D}}) < B. Using smooth approximation, we can find a smoothly immersed disc D′ spanning ℓ(γ) with ∫_{D′} ω < ε and A(D′) < B. It follows that for i sufficiently large, there is a smoothly immersed disc D′_{i} spanning ℓ_{i}(γ) with ∫_{D′_{i}} ω < 2ε and A(D′_{i}) < B. The result follows. q.e.d.

Lemma 5.16. If μ ∈ T (Σ) is a critical point for the function Ẽ, then the weakly lagrangian map ℓ_{μ} is weakly conformal.

Proof. A variation of the domain shows that the Hopf differential is holomorphic and hence smooth. The result then follows from an argument in [SU1]. q.e.d.

Theorem 5.17. Suppose Σ is a surface of genus g ≥ 2 and ϕ : Σ → N is a lagrangian map that is non-collapsible. Then there is a conformal structure on Σ and a weakly conformal, weakly lagrangian map ℓ ∈ L̃_{ϕ} that minimizes area among all maps in L̃_{ϕ}.

Proof. Let ν be the conformal structure on Σ for the metric induced by ϕ. Then E(ϕ, ν) = A(ϕ) < C. In particular, Ẽ_{ν} < C. To prove the theorem it suffices to show, by Proposition 5.4, that a minimum of Ẽ is attained in the interior of T (Σ). Let μ_{i} ∈ T (Σ) be a minimizing sequence for Ẽ. We can suppose that Ẽ_{μ_{i}} < C. Then there is a subsequence, still denoted μ_{i}, and a sequence of elements τ_{i} ∈ Mod(Σ) such that ξ_{i} = τ_{i}(μ_{i}) either converges to an element in the interior of T (Σ) or the marked surface Σ_{ξ_{i}} representing ξ_{i} converges to a surface with nodes [A]. Let ℓ_{i} be a minimizing map in L̃_{ϕ·τ_{i}^{−1}} for the conformal

structure ξ_{i}. Clearly E(ℓ_{i}, ξ_{i}) = Ẽ_{μ_{i}} < C. Therefore since a nontrivial simple closed curve in Σ either maps to a nontrivial curve in N or, if not, to a curve with period_{C} bounded below, by Theorem 5.15 we have lim_{i→∞} ξ_{i} = ξ ∈ T (Σ). Choosing a subsequence, we can suppose that {ℓ_{i}} converges weakly in W^{1,2} to ℓ and lim inf_{i→∞} Ẽ_{μ_{i}} = E(ℓ, ξ). It follows from Lemma 5.16 that ℓ is weakly conformal. q.e.d.

We next consider the case where genus (Σ) = 1. Let T^{2} be a closed surface of genus one. We include the period condition in the following definition however it will not be necessary in our applications. Fix δ > 0. Suppose that ϕ : T^{2} → N is a piecewise C^{1} lagrangian map such that if [γ] is a generator of π_{1}(T^{2}, p) that lies in the kernel of ϕ_{∗} : π_{1}(T^{2}, p) → π_{1}(N, ϕ(p)) then period_{C}(ϕ, [γ]) ≥ δ > 0. We require , in addition, that C satisfy 2A(ϕ) < C and that 2C is a regular value of the energy functional E(S^{2}, N) on piecewise C^{1} maps S^{2} → N. We call such a lagrangian map non-collapsible. Note that every generator of π_{1}(T^{2}, p) can be represented by a simple closed curve.

In analogy to the above definitions we define:

L̃_{ϕ} = {ℓ ∈ W^{1,2}(T^{2}, N) : ℓ is weakly lagrangian,

ℓ has the same action on π_{1} as ϕ and

the period_{C} invariants of ℓ and ϕ are equal}.

The proof of the following theorem is also a variation of the work of [SU2] and is left to the reader.

Theorem 5.18 Suppose T^{2} is a torus and ϕ : T^{2} → N is a noncollapsible lagrangian map. Then there is a conformal structure on T^{2} and a weakly conformal, weakly lagrangian map ℓ ∈ L̃_{ϕ} that minimizes area among all maps in L̃_{ϕ}.

Theorem 5.19. Let Σ be a surface of genus g ≥ 1 and ϕ : Σ → N be a lagrangian map that is non-collapsible. Let α be a lagrangian homology class. Suppose that there are maps f ∈ L̃_{ϕ} such that [f] = α. Then there is a conformal structure on Σ and weakly conformal, weakly lagrangian maps ℓ : Σ → N and s_{i} : S^{2} → N, i = 1, …, ν, such that:

(1) Each map s_{i} minimizes area among lagrangian maps in some free homotopy class.

(2) The map ℓ is non-collapsible and minimizes area among all maps in L̃_{ϕ}.

(3) [ℓ] + ∑_{i}^{ν}[s_{i}] = α.

Proof. The result follows by applying the argument in the proofs of Propositions 5.5 and 5.7 to an area minimizing sequence of maps in L̃_{ϕ} that represent α. q.e.d.

Minimizing in a homology class

Theorem 5.20. Suppose that α is a lagrangian class in H_{2}(N; ℤ). Then α can be represented by a finite number of piecewise C^{1} lagrangian maps of surfaces into N each of which is non-collapsible.

Proof. Without loss of generality, we assume that α cannot be written as a sum of two nontrivial classes each of which is lagrangian. We represent α by a piecewise C^{1} lagrangian map ℓ : Σ → N. We can suppose that ℓ is an immersion except in the neighborhood of finitely many points and, by perturbation, we can suppose that, ℓ has, at worst, double points. We take the surface Σ to have minimal genus among such surfaces representing α. Since α is a lagrangian class,

∫_{Σ} ℓ^{∗}ω = 0. (5.1)

If γ is a simple closed curve on Σ that does not separate and for which ℓ(γ) is trivial in π_{1}(N) then ℓ(γ) can be contracted without changing the homology class of ℓ(Σ) and preserving (5.1). Thus we can assume that all such curves are nontrivial. If for every simple closed curve γ on Σ which separates we have that ℓ_{∗}([γ]) ≠ 1 then the theorem is proved. So suppose that γ be a simple closed curve on Σ which separates and for which ℓ_{∗}([γ]) is trivial in π_{1}(N, x). We can also suppose ℓ(γ) is imbedded. ℓ(γ) bounds an immersed disc D in N. If this disc is lagrangian then ℓ(Σ) can be cut along ℓ(γ) into two lagrangian surfaces, contradicting the indecomposibility of α. If ∫_{D} ω = 0 then D is homotopic rel ∂D = ℓ(γ) to an immersed lagrangian disc. Thus we can suppose that for any such γ there are no immersed discs D in N spanning ℓ(γ) such that ∫_{D} ω = 0. In particular, period_{C}(ℓ, [γ]) ≠ 0, where we choose C > 0 to satisfy 2A(ℓ(Σ)) < C and such that 2C is a regular value of the energy functional on piesewise C^{1} maps S^{2} → N.

There can be many homotopy classes in π_{1}(Σ, p) that are represented by simple closed separating curves. Let [γ_{1}] and [γ_{2}] be two such classes and let γ_{1} and γ_{2} be simple closed curves that are representatives. Recalling the notation introduced in Lemma 5.10 , for i = 1, 2, let

S(ℓ, [γ_{i}], C) = ∪_{γ∈[γ_{i}]}S(ℓ, γ, C).

Let D_{i} be a disc in N with A(D_{i}) < C and ∂D_{i} = ℓ(γ_{i}). Since γ_{1} and γ_{2} are both separating curves they bound a region U ⊂ Σ. In N construct the cycle

Σ_{D_{1},D_{2}} = D_{1} ∪_{ℓ(γ_{1})} ℓ(U) ∪_{ℓ(γ_{2})} D_{2}.

Since ℓ is lagrangian,

ω([Σ_{D_{1},D_{2}}]) = ∫_{D_{1}} ω − ∫_{D_{2}} ω.

Making different choices of spanning discs D_{1} and D_{2} we construct different cycles and get different numbers after pairing with [ω]. However note that A(Σ_{D_{1},D_{2}}) < 2C + A(ℓ(Σ)). Using Lemma 5.3 there are at most finitely many homology classes in H_{2}(N, ℤ) that can be represented by cycles with area less than 2C + A(ℓ(Σ)). Pairing these homology classes with [ω] we get finitely many scalars T = {t_{1}, …, t_{λ}}. It follows that each element of S(ℓ, γ_{1}, C) differs from one of S(ℓ, γ_{2}, C) by addition or subtraction by some element of T. This is true for each representative γ_{1} of [γ_{1}] and γ_{2} of [γ_{2}] and therefore it is true for the elements of the sets S(ℓ, [γ_{1}], C) and S(ℓ, [γ_{2}], C). Fix a homotopy class [γ_{0}] ∈ π_{1}(Σ, p) that can be represented by a simple closed separating curve. Let [γ] ∈ π_{1}(Σ, p) be any other such class. Then the elements of S(ℓ, [γ], C) can be obtained from the elements of S(ℓ, [γ_{0}], C) by addition or subtraction by an element of T. Hence the union of the sets S(ℓ, [γ], C), over all classes [γ] ∈ π_{1}(Σ, p) that can be represented by simple closed separating curves, has finitely many elements. Denote the infimum by δ. If δ = 0 then ℓ(Σ) can be cut into two lagrangian surfaces, contradicting the indecomposibility of α. This proves the theorem. q.e.d.

Let α be a lagrangian homology class. By the previous result we can represent α by a sum of non-collapsible lagrangian surfaces and lagrangian 2-spheres. Applying the minimization arguments of this section to each map in this decomposition we have:

Theorem 5.21. Let α ∈ H_{2}(N; ℤ) be a lagrangian homology class. Then there exist Riemann surfaces Σ_{1}, …, Σ_{μ} and hamiltonian stationary, weakly conformal, weakly lagrangian maps ℓ_{j} : Σ_{j} → N, j = 1, …, μ,and s_{i} : S^{2} → N, i = 1, …, ν such that:

(1) Each map s_{i} minimizes area among lagrangian maps in a free homotopy class.

(2) Each map ℓ_{j} minimizes area among lagrangian maps Σ_{j} → N with fixed induced action on π_{1} and fixed periods.

(3) The maps ℓ_{j} : Σ_{j} → N are non-collapsible.

(4) ∑_{j}^{μ}[ℓ_{j}] + ∑_{i}^{ν}[s_{i}] = α, where [ℓ_{j}], [s_{i}] ∈ H_{2}(N; ℤ).

6. Second variation formula

In this section we derive the second variation formula for immersed lagrangian submanifolds that are hamiltonian stationary. However, to begin we digress to state and prove the general second variation formula for smoothly immersed submanifolds possibly with boundary. We do not assume that the variation vector field is normal or that the immersed submanifold is stationary.

Let M be a Riemannian manifold of dimension m with metric g. Let ℓ : Σ → M be a smooth immersion, where Σ is a compact manifold of dimension n with boundary (possibly empty). A smooth mapping

L : (−ε, ε) × Σ → M

satisfying:

(i) Each map ℓ_{t} = L(t, −) : Σ → M is a smooth immersion;

(ii) ℓ_{0} = ℓ;

will be called a smooth variation of ℓ. Let \frac{∂}{∂t} denote the vector field along the (−ε, ε) factor; and denote X = L_{∗}(\frac{∂}{∂t})|_{t=0}. X is a vector field along ℓ(Σ) called the variation vector field.

Let p ∈ Σ. With respect to the metric induced on Σ by ℓ, choose an orthonormal frame {e_{1} ⋯ e_{n}} to satisfy:

∇_{e_{i}}^{Σ}e_{j}(p) = 0 for all i, j,

where ∇^{Σ} denotes the Levi-Civita connection on Σ for the metric induced by ℓ. For each t ∈ (−ε, ε), the vector fields {ℓ_{t∗}e_{1}, …, ℓ_{t∗}e_{n}} give a framing along ℓ_{t}(Σ). Note that the metric on Σ induced by ℓ_{t} can be written

g_{ij}(t) = ⟨ℓ_{t∗}e_{i}, ℓ_{t∗}e_{j}⟩, 1 ≤ i, j ≤ n.

For brevity of notation we will write

e_{i}(t) = ℓ_{t∗}e_{i}, 1 ≤ i ≤ n.

Let dv_{t} denote the volume form of the metric induced by ℓ_{t}. Set

V (t) = ∫_{Σ} dv_{t}.

Let B denote the second fundamental form on ℓ(Σ). B is a symmetric bilinear form on T Σ with values in the normal bundle. R will denote the curvature tensor of g on M and H the mean curvature vector.

Theorem 6.1 (Second Variation Formula). Let ℓ : Σ → M be a smoothly immersed submanifold, possibly with boundary. Let L : (−ε, ε) × Σ → M be a smooth variation of ℓ with variation vector field X. Then

\frac{d^{2}V}{dt^{2}} |_{t=0} = ∫_{Σ} [∑_{i} |(∇_{e_{i}}X)^{⊥}|^{2} + ∑_{i}⟨R_{e_{i},X}e_{i}, X⟩ − ⟨∇_{X}X, H⟩

− ∑_{i,j}⟨∇_{e_{j}}X, e_{i}⟩⟨∇_{e_{i}}X, e_{j}⟩ + ∑_{i,j}⟨∇_{e_{i}}X, e_{i}⟩⟨∇_{e_{j}}X, e_{j}⟩] dv_{0}

− ∫_{∂Σ}⟨∇_{X}X, ν⟩dv_{∂Σ}

where {e_{1}, …, e_{n}} is an orthonormal frame on ℓ_{0}(Σ) and ν is the inward pointing conormal vector field along ℓ_{0}(∂Σ).

Proof. The well-known computation of the first variation formula shows that:

\frac{d}{dt}dv_{t} = ∑_{i,j} g^{ij}(t)⟨∇_{e_{i}}X, e_{j}⟩dv_{t},

where g^{ij}(t) is the inverse matrix of the metric g_{ij}(t) = ⟨e_{i}(t), e_{j}(t)⟩. It follows that

\frac{d^{2}V}{dt^{2}} |_{t=0} = \frac{d}{dt} ∫_{Σ} ∑_{i,j} g^{ij}(t)⟨∇_{e_{i}}X, e_{j}⟩dv_{t}|_{t=0}

= ∫_{Σ} \frac{d}{dt} [∑_{i,j} g^{ij}(t)⟨∇_{e_{i}}X, e_{j}⟩]|_{t=0} dv_{0}

+ ∑_{i}⟨∇_{e_{i}}X, e_{i}⟩\frac{d}{dt}dv_{t} |_{t=0} .

Since g^{ij}(0) = δ_{ij},

\frac{d}{dt}dv_{t}|_{t=0} = ∑_{i}⟨∇_{e_{i}}X, e_{i}⟩dv_{0}.

Using [X, e_{i}] = 0, i = 1, …, n we have

\frac{dg^{ij}}{dt} (0) = −\frac{dg_{ij}}{dt} (0) = −X⟨e_{i}, e_{j}⟩ = −⟨∇_{e_{i}}X, e_{j}⟩ − ⟨e_{i}, ∇_{e_{j}}X⟩.

minimizing area among lagrangian surfaces 63

Thus,

\frac{d^{2}V}{dt^{2}} |_{t=0} = ∫_{Σ} [∑_{ij}(−⟨∇_{e_{i}}X, e_{j}⟩ − ⟨e_{i}, ∇_{e_{j}}X⟩)⟨∇_{e_{i}}X, e_{j}⟩

+ ∑_{i}⟨∇_{X}∇_{e_{i}}X, e_{i}⟩ + ∑_{i}⟨∇_{e_{i}}X, ∇_{X}e_{i}⟩

+ ∑_{i,j}⟨∇_{e_{i}}X, e_{i}⟩⟨∇_{e_{j}}X, e_{j}⟩] dv_{0}.

Note that

⟨∇_{X}∇_{e_{i}}X, e_{i}⟩ = ⟨R_{X,e_{i}}X, e_{i}⟩ + ⟨∇_{e_{i}}∇_{X}X, e_{i}⟩.

Thus at a point p ∈ M where ∇_{e_{i}}e_{j}(p) = 0 we have

∑_{i}⟨∇_{e_{i}}∇_{X}X, e_{i}⟩ = ∑_{i} e_{i}⟨∇_{X}X, e_{i}⟩ − ∑_{i}⟨∇_{X}X, ∇_{e_{i}}e_{i}⟩
= ∑_{i} e_{i}⟨∇_{X}X, e_{i}⟩ − ⟨∇_{X}X, H⟩,

where H is the mean curvature. It follows that

\frac{d^{2}V}{dt^{2}} |_{t=0} = ∫_{Σ} [− ∑_{i} |(∇_{e_{i}}X)^{⊤}|^{2} + ∑_{i} |∇_{e_{i}}X|^{2} + ∑_{i}⟨R_{X,e_{i}}X, e_{i}⟩

− ⟨∇_{X}X, H⟩ − ∑_{i,j}⟨e_{i}, ∇_{e_{j}}X⟩⟨e_{j}∇_{e_{i}}X⟩

+ ∑_{i,j}⟨∇_{e_{i}}X, e_{i}⟩⟨∇_{e_{j}}X, e_{j}⟩ + ∑_{i} e_{i}⟨∇_{X}X, e_{i}⟩] dv_{0}.

The result now follows by applying the divergence theorem to give:

∫_{Σ} ∑_{i} e_{i}⟨∇_{X}X, e_{i}⟩dv_{0} = − ∫_{∂Σ}⟨∇_{X}X, ν⟩dv_{0},

where ν is the inward pointing conormal. q.e.d.

We now return to the lagrangian case. Let M be a Kähler manifold of complex dimension n with Kähler form ω, Kähler metric g and complex structure J. Let ℓ : Σ → M be a lagrangian immersion, where Σ is a compact manifold with boundary (possibly empty). A smooth mapping

L : (−ε, ε) × Σ → M

satisfying:

(i) Each map ℓ_{t} = L(t, −) : Σ → M is a lagrangian immersion;

(ii) ℓ_{0} = ℓ;

will be called a smooth lagrangian variation of ℓ. The variation vector field X = L_{∗}(\frac{∂}{∂t})|_{t=0} will be called a lagrangian variation. Analogous terms will be used in the hamiltonian case.

Lemma 6.2. Suppose Σ is a hamiltonian stationary submanifold. If X denotes the variational vector field along Σ of a hamiltonian variation then

∫_{Σ}(⟨H, ∇_{X}X⟩ − ⟨X, ∇_{JH}JX⟩) dvol = 0

where H denotes the mean curvature vector field along Σ and ∇ is the Levi-Civita connection of g on M.

Proof. Since X is hamiltonian there is a smooth function h such that JX = ∇h. Define a 1-form σ on Σ by:

σ(V) = ⟨J(∇_{X}X), V ⟩ − ⟨X, ∇_{V} JX⟩,

for V ∈ T (Σ). Then

σ(V) = ⟨J(∇_{X}X), V ⟩ − ⟨X, ∇_{V} JX⟩
= X⟨JX, V ⟩ − ⟨JX, ∇_{X}V ⟩ + ⟨JX, ∇_{V} X⟩
= X⟨JX, V ⟩ + ⟨JX, [V, X]⟩.
= X⟨∇h, V ⟩ + ⟨∇h, [V, X]⟩.
= X(V (h)) + [V, X](h).
= V (X(h)).

It follows that σ = d(X(h))_{|_{Σ}}, in particular, σ is an exact 1-form. Using σ in the first variational formula we get:

∫_{Σ}(⟨H, ∇_{X}X⟩ − ⟨X, ∇_{JH}JX⟩)dvol = 0.

This proves the lemma. q.e.d.

Using a slightly different argument the following result can be proved. However since we won’t require it we leave the proof to the reader.

Lemma 6.3. Suppose Σ is a lagrangian stationary submanifold. If X denotes the variational vector field along Σ of a lagrangian variation then

∫_{Σ}(⟨H, ∇_{X}X⟩ − ⟨X, ∇_{JH}JX⟩) dvol = 0

where H denotes the mean curvature vector field along Σ and ∇ is the Levi-Civita connection of g on M.

The next theorem is due to Y. G. Oh [O] who derived it, and applied it to prove the stability of certain hamiltonian stationary submanifolds. We include the proof here for the convenience of the reader.

Theorem 6.4 (Y.G. Oh). Let ℓ : Σ → M be a hamiltonian stationary submanifold. Let L : (−ε, ε) × Σ → M be a smooth hamiltonian variation of ℓ with variation vector field X, that leaves the boundary fixed. Denote by X^{⊥} the normal part of X. Then

\frac{d^{2}V}{dt^{2}} |_{t=0} = ∫_{Σ} [∑_{i} |∇_{e_{i}}^{Σ}JX^{⊥}|^{2} + ∑_{i}⟨R_{e_{i},X^{⊥}}e_{i}, X^{⊥}⟩ + ⟨X^{⊥}, H⟩^{2}

− ∑_{i,j}⟨X^{⊥}, B_{e_{i},e_{j}}⟩^{2} − ⟨X^{⊥}, B_{JH,JX^{⊥}}⟩] dv_{0}

= ∫_{Σ} [|δσ|^{2} − Ric(X^{⊥}, X^{⊥}) + ⟨X^{⊥}, H⟩^{2}

− 2⟨X^{⊥}, B_{JH,JX^{⊥}}⟩]dv_{0}

where σ = X^{⊥} ω is a closed 1-form, δ is the adjoint of d on Σ and Ric is the Ricci curvature on N.

Proof. The first equality follows from the lemma. To see the second equality we note that since Σ is lagrangian, for vector fields X, Y, Z on Σ, the second fundamental form satisfies the symmetry:

⟨B_{X,Y} , JZ⟩ = ⟨B_{X,Z}, JY ⟩.

Using this and the Gauss equation we have:

(6.1) ∑_{i}⟨R_{e_{i},JX^{⊥}}e_{i}, JX^{⊥}⟩ = ∑_{i}⟨R_{e_{i},JX^{⊥}}e_{i}, JX^{⊥}⟩ − ⟨X^{⊥}, B_{JH,JX^{⊥}}⟩

+ ∑_{i,j}⟨X^{⊥}, B_{e_{i},e_{j}}⟩^{2},

where R denotes the curvature tensor on Σ. The Weitzenböck formula relating the Hodge and covariant laplacians on 1-forms on Σ gives:

Δ_{h}σ = −∇^{2}JX^{⊥} + Ric(JX^{⊥}, JX^{⊥}), (6.2)

where Δ_{h} is the Hodge laplacian and Ric denotes the Ricci curvature on Σ. Combining (6.1), (6.2) and integrating by parts gives the result.

q.e.d.

7. Two dimensional stationary lagrangian cones

In this section we give a complete description of the stationary lagrangian cones in ℝ^{4}. We apply the second variation formula to study their stability properties and conclude with a discussion of their minimization properties.

To begin consider the unit sphere in ℝ^{2n}, endowed with its induced CR structure as well as its induced metric. To describe this CR structure, let p⃗ denote the position vector at a point p ∈ 𝕊^{2n−1}. Then T = Jp⃗ is a tangent vector field, and the orthogonal complement Π_{p} = T^{⊥} in the sphere is a J-invariant 2n − 2 dimensional subspace. This is the contact (CR) distribution in the sphere. If we consider a cone C(Γ) over an n − 1 dimensional submanifold Γ of the unit sphere, then we see that C(Γ) is lagrangian if and only if Γ is legendrian in the sphere. This is true because the tangent plane to the cone at points of Γ is spanned by the tangent space to Γ together with the position vector, and the legendrian condition guarantees that the position vector is taken by J to a vector orthogonal to the cone. In particular, we see that a two dimensional cone is lagrangian if and only if Γ is a legendrian curve.

We now consider a curve γ in 𝕊^{3}. We introduce complex coordinates z_{1}, z_{2} and we write γ(s) = (γ_{1}(s), γ_{2}(s)) where γ_{j} are l periodic complex valued functions and s is an arclength parameter with l = length(γ). The condition that the cone over γ be lagrangian is that the 2×2 matrix with columns γ and γ̇ is a unitary matrix. We have

e^{iβ} = γ_{1}γ̇_{2} − γ_{2}γ̇_{1}

where β is the lagrangian angle. In order that the cone be hamiltonian stationary, β must be a harmonic function. On the other hand β is homogeneous of degree 0 on the cone. It follows that β is a linear function of s which we can take to be 2as for a real constant a. These conditions imply that

γ̇_{1} = −e^{i2as}γ_{2}, γ̇_{2} = e^{i2as}γ_{1}. (7.1)

If we differentiate the first equation and substitute in from the second we get

γ̈_{1} = i2aγ̇_{1} − γ_{1}.

Thus we have

γ_{1}(s) = c_{1}e^{i(a+\sqrt{a^{2}+1})s} + c_{2}e^{i(a−\sqrt{a^{2}+1})s}

for complex constants c_{1}, c_{2}. It follows that

γ_{2}(s) = i(a + \sqrt{a^{2} + 1})c_{1}e^{i(a−\sqrt{a^{2}+1})s} + i(a − \sqrt{a^{2} + 1})c_{2}e^{i(a+\sqrt{a^{2}+1})s}.

Since γ is to be a curve on the unit sphere, we must have the condition

|γ_{1}|^{2} + |γ_{2}|^{2} = |c_{1}|^{2}(1 + (a + \sqrt{a^{2} + 1})^{2}) + |c_{2}|^{2}(1 + (a − \sqrt{a^{2} + 1})^{2}) = 1.

In order that γ be a closed curve, it must be true that a + \sqrt{a^{2} + 1} and a − \sqrt{a^{2} + 1} are rationally related. One can check that this condition is equivalent to saying that there are relatively prime positive integers p, q such that a + \sqrt{a^{2} + 1} = \sqrt{p/q} and a − \sqrt{a^{2} + 1} = −\sqrt{q/p}. We then have that the length of the curve satisfies l = 2π\sqrt{pq}, and if we set θ = s/\sqrt{pq}, then we have γ(θ) is a 2π periodic parametrization. Furthermore we have β = (\sqrt{p/q} − \sqrt{q/p})s = (p − q)θ. Thus we find that β has a period around γ which is equal to 2π(p − q). Applying a unitary transformation we can take c_{1} to be real and c_{2} = 0, so we see that γ is unitarily equivalent to the curve

γ(s) = \frac{1}{\sqrt{p + q}} (\sqrt{q}e^{i\sqrt{\frac{p}{q}} s}, i\sqrt{p}e^{−i\sqrt{\frac{q}{p}} s}) , (7.2)

where 0 ≤ s ≤ 2π\sqrt{pq}. We note that these curves are not great circles except in the special case in which p = q = 1. In general they lie on Clifford tori in 𝕊^{3}. When both p > 1 and q > 1 the curves are knotted. If either p = 1 or q = 1 they are unknotted. Thus we have shown:

Theorem 7.1. The lagrangian cones C_{γ} in ℝ^{4} that are hamiltonian stationary are the cones over the curves γ of (7.2). They are parameterized by a pair of relatively prime positive integers (p, q).

The second variation formula takes a particularly simple form when applied to variations of the cone, C_{γ}, over γ whose variation vector field X is hamiltonian, i.e., when X = J∇f for f ∈ C_{c}^{∞}(C_{γ}).

Proposition 7.2. Suppose that the cone C_{γ} over γ is lagrangian and hamiltonian stationary. Let X be a compactly supported hamiltonian variation, X = J∇f, f ∈ C_{c}^{∞}(C_{γ}) that leaves the cone vertex fixed. Then the second variation formula with variation vector field X is:

\frac{d^{2}V}{dt^{2}} |_{t=0} = ∫_{C_{γ}} [(Δf)^{2} − r^{−4} \frac{(q − p)^{2}}{pq} (f_{s})^{2}] rdrds (7.3)

where Δ is the Laplacian on C_{γ}.

Proof. Let r denote the radial coordinate on C_{γ}. With respect to the coordinates r and s the induced metric on C_{γ} has the form

dσ^{2} = dr^{2} + \frac{1}{r^{2}} ds^{2}.

Set

e_{1} = γ, e_{2} = γ̇.

Then {e_{1}, e_{2}} is an orthonormal frame on C_{γ}. With respect to this frame an easy computation shows that the second fundamental form satisfies

B_{e_{1},e_{1}} = B_{e_{1},e_{2}} = 0, B_{e_{2},e_{2}} = \frac{1}{r}(γ̈ + γ). (7.4)

In particular, H = \frac{1}{r}(γ̈ + γ). From (7.4) it follows immediately that

⟨H, X⟩^{2} − ∑_{i,j}⟨X, B_{e_{i},e_{j}}⟩^{2} = 0.

To compute ⟨X, ∇_{JH}JX⟩, we note that, from the explicit form of γ (7.2), we have:

JH = \frac{1}{r} \frac{q − p}{\sqrt{pq}}γ̇ = \frac{1}{r} \frac{q − p}{\sqrt{pq}}e_{2}.

Clearly,

JX = −∇f = −f_{r}e_{1} − \frac{1}{r}f_{s}e_{2}.

Thus,

⟨X, ∇_{JH}JX⟩ = ⟨X, B_{JH,JX}⟩
= ⟨X, −\frac{1}{r^{2}} \frac{(q − p)}{\sqrt{pq}}f_{s}B_{e_{2},e_{2}}⟩

= −\frac{1}{r^{2}} \frac{(q − p)}{\sqrt{pq}}f_{s}⟨JX, JH⟩

= \frac{1}{r^{4}} \frac{(q − p)^{2}}{pq} (f_{s})^{2}.

The result follows. q.e.d.

Remark 7.1. The coordinate functions are easily seen to be Jacobi fields.

We next apply Proposition 7.2 to study the stability properties of the lagrangian cones for various pairs p, q.

Proposition 7.3. The stationary lagrangian cones with |p − q| > 1 are strictly unstable. This is true for lagrangian variations fixing a neighborhood of the cone vertex.

Proof. Choose a positive integer ℓ satisfying

ℓ(−|p − q| + ℓ) < pq < ℓ(|p − q| + ℓ). (7.5)

Set

f(r, s) = ζ(r) cos \frac{ℓs}{\sqrt{pq}} 0 ≤ s ≤ 2π\sqrt{pq}

where ζ(r) ∈ C_{c}^{∞}(ℝ_{+}) will be specified later. Then we have,

Δf = Δζ cos \frac{ℓs}{\sqrt{pq}} − r^{−2}ζ \frac{ℓ^{2}}{pq} cos \frac{ℓs}{\sqrt{pq}}

= (ζ′′ + \frac{ζ′}{r} − ζ \frac{ℓ^{2}}{pq} r^{−2}) cos \frac{ℓs}{\sqrt{pq}} .

Using f in the second variation formula (7.3) we have:

(7.6)  \frac{d^{2}V}{dt^{2}} |_{t=0} = ∫ [ (ζ′′ + \frac{ζ′}{r} − ζ \frac{ℓ^{2}}{pq} r^{−2})^{2} cos^{2} \frac{ℓs}{\sqrt{pq}}

− r^{−4} \frac{(p − q)^{2}}{pq} ζ^{2} \frac{ℓ^{2}}{pq} sin^{2} \frac{ℓs}{\sqrt{pq}} ]rdrds.

70 r. schoen & j. wolfson

Set

g(r, s) = ζ(r) sin \frac{ℓs}{\sqrt{pq}} 0 ≤ s ≤ 2π\sqrt{pq}

where ζ(r) is as above. Using g in the second variation formula (7.3) we have:

(7.7)  \frac{d^{2}V}{dt^{2}} |_{t=0} = ∫ [ (ζ′′ + \frac{ζ′}{r} − ζ \frac{ℓ^{2}}{pq} r^{−2})^{2} sin^{2} \frac{ℓs}{\sqrt{pq}}

− r^{−4} \frac{(p − q)^{2}}{pq} ζ^{2} \frac{ℓ^{2}}{pq} cos^{2} \frac{ℓs}{\sqrt{pq}} ]rdrds.

We wish to show that for suitable choice of ζ one of (7.6), (7.7) is negative. To do this add the integrals to give:

2π\sqrt{pq} ∫ [(ζ′′ + \frac{ζ′}{r} − ζ \frac{ℓ^{2}}{pq} r^{−2})^{2} − r^{−4} \frac{(p − q)^{2}}{(pq)^{2}} ℓ^{2}ζ^{2}] rdr. (7.8)

Next choose ε, 0 < ε < 1 and define ζ:

ζ(r) = \{ \begin{matrix}δ(r) & 0 ≤ r ≤ ε \\ r & ε ≤ r ≤ 1 \\ η(r) & r ≥ 1 \end{matrix}

where η satisfies:

(i) η has support on [1, 2]; (ii) η(1) = 1, η′(1) = 1, η′′(1) = 0; (iii) 0 ≤ η ≤ c, |η′| ≤ c, |η′′| ≤ c for some constant c > 0; and where δ satisfies:

(i) δ has support on [\frac{ε}{2}, ε]; (ii) δ(ε) = ε, δ′(ε) = 1, δ′′(ε) = 0; (iii) 0 ≤ δ ≤ ε, |δ′| ≤ 4, |δ′′| ≤ \frac{4}{ε}.

Thus (7.8) becomes three integrals:

2π\sqrt{pq} ∫ ε_{\frac{ε}{2}} [(δ′′ + \frac{δ′}{r} − δ \frac{ℓ^{2}}{pq} r^{−2})^{2} − r^{−4} \frac{(q − p)^{2}}{(pq)^{2}} ℓ^{2}δ^{2}] rdr

+ 2π\sqrt{pq} ∫_{ε}^{1} [(pq − ℓ^{2})^{2} − ℓ^{2}(p − q)^{2}] \frac{1}{(pq)^{2}} \frac{dr}{r}

+ 2π\sqrt{pq} ∫_{1}^{2} [(η′′ + \frac{η′}{r} − η \frac{ℓ^{2}}{pq} r^{−2})^{2} − r^{−4} \frac{(q − p)^{2}}{(pq)^{2}} ℓ^{2}η^{2}] rdr.

The absolute value of the first integral is bounded by

C ∫ ε_{\frac{ε}{2}} \frac{dr}{ε} = \frac{C}{2}

where C depends on ℓ, p, q but is independent of ε. The third integral is clearly bounded. The second integral equals

−2π(pq)^{− \frac{3}{2}} [(pq − ℓ^{2})^{2} − ℓ^{2}(p − q)^{2}] ln ε.

By our choice of ℓ this expression is negative and so, for ε sufficiently small, the sum of all three integrals is negative. We conclude that one (or both) of (7.6), (7.7) is negative. q.e.d.

Suppose now that |p − q| = 1. Without loss of generality we can suppose that q = p + 1.

Proposition 7.4. The stationary lagrangian cones with |p − q| = 1 are strictly stable for lagrangian variations fixing the cone vertex.

Proof. Without loss of generality we can assume that the lagrangian variation is a normal hamiltonian variation, i.e., has the form X = J∇f for some f ∈ C_{c}^{∞}(C_{γ} \ {0}) satisfying f(0) = f_{r}(0) = 0. We first argue that the second variation is positive for variations of the form X = J∇f_{ℓ} or X = J∇g_{ℓ} where,

f_{ℓ}(r, s) = ζ(r) cos \frac{ℓs}{\sqrt{pq}} , 0 ≤ s ≤ 2π\sqrt{pq},

or

g_{ℓ}(r, s) = ζ(r) sin \frac{ℓs}{\sqrt{pq}} , 0 ≤ s ≤ 2π\sqrt{pq}.

for any positive integer ℓ. ζ(r) is assumed to have compact support and to satisfy:

ζ(0) = ζ′(0) = 0

(so that X(0) = 0). The computation in the proof of Proposition 7.3 shows that the second variation for both f_{ℓ} and g_{ℓ} reduces to a positive multiple of:

∫_{0}^{∞} [(ζ′′ + \frac{ζ′}{r} − ζ \frac{ℓ^{2}}{pq} r^{−2})^{2} − r^{−4} \frac{1}{(pq)^{2}} ℓ^{2}ζ^{2}] rdr. (7.9)

Make the substitutions,

r = e^{t}, ζ(r) = e^{t}ρ(t).

The boundary conditions on ρ are:

ρ(t) → 0, ρ′(t) → 0, as t → ±∞.

Using the same argument as in the proof of Proposition 7.4 the integral (7.9) becomes:

∫_{−∞}^{∞} [(ρ′′)^{2}+ (2 + \frac{2p}{p + 1})(ρ′)^{2}+ \frac{1}{(p(p + 1))^{2}} [(p(p + 1) − ℓ^{2})^{2}− ℓ^{2}]ρ^{2}] dt.

Since,

[(p(p + 1) − ℓ^{2})^{2} − ℓ^{2}] ≥ 0,

for any ℓ ∈ ℤ_{+}, this integral is positive for any ρ.

If f ∈ C_{c}^{∞}(C_{γ} \ {0}) satisfies f(0) = f_{r}(0) = 0 then f has a Fourier series expansion:

f(r, s) = ∑_{ℓ=0}^{∞} (ζ_{ℓ}(r) cos \frac{ℓs}{\sqrt{pq}} + η_{ℓ}(r) sin \frac{ℓs}{\sqrt{pq}} ) ,

with

ζ_{ℓ}(0) = ζ′_{ℓ}(0) = η_{ℓ}(0) = η′_{ℓ}(0) = 0, ζ, η ∈ C_{c}^{∞}((0, ∞)).

It follows using the L^{2} perpendicularity of sine and cosine and the above computation that

dV (J∇f, J∇f) > 0.

The result follows. q.e.d.

The situation for multiply covered cones is markedly different.

Proposition 7.5. The multiply covered stationary lagrangian cones for any p, q are strictly unstable. This is true for lagrangian variations fixing a neighborhood of the cone vertex.

Proof. The k times covered (p, q) cones are parameterized by:

\frac{1}{\sqrt{p + q}} (r\sqrt{q}e^{i\sqrt{\frac{p}{q}} s}, ir\sqrt{p}e^{−i\sqrt{\frac{q}{p}} s}) , (7.10)

where 0 ≤ s ≤ 2kπ\sqrt{pq}, 0 ≤ r < ∞ and k > 1 is an integer. Without loss of generality we can suppose q > p. Put:

ℓ = p + \frac{1}{k} . (7.11)

Set:

f(r, s) = ζ(r) cos \frac{ℓs}{\sqrt{pq}} 0 ≤ s ≤ 2kπ\sqrt{pq}

and

g(r, s) = ζ(r) sin \frac{ℓs}{\sqrt{pq}} 0 ≤ s ≤ 2kπ\sqrt{pq}

where ζ(r) ∈ C_{c}^{∞}(ℝ_{+}) is as given in the proof of Proposition 7.3. Then both f and g are compactly supported hamiltonians on the k covered (p, q) cone. Using the argument in the proof of Proposition 7.3 it suffices to show

(pq − ℓ^{2})^{2} − ℓ^{2}(p − q)^{2} < 0 (7.12)

to conclude that the second variation with respect to f or g (or both) is negative. Using ℓ = p + \frac{1}{k} and q > p this follows easily. q.e.d.

We now show that none of the cones are minimizing if we allow comparisons which are lagrangian and nonorientable. A convenient way to do this is to use the following generalization of Allcock’s theorem [Al] due to Weiyang Qiu [Q]. The idea here is that Allcock’s method can be extended to give a lagrangian homotopy (with controlled area) of any curve Γ to a twice covered circle in a complex plane with the same enclosed symplectic area as that of Γ. This then represents a lagrangian Möbius band.

Lemma 7.6 (Qiu). Let Γ be any rectifiable curve in ℝ^{4}. There exists a lagrangian Möbius band Σ with boundary Γ and such that A(Σ) ≤ cL(Γ)^{2} for a fixed constant c.

The same argument as in the proof of Lemma 4.8 now shows the following:

Lemma 7.7. Suppose that ℓ_{0}, ℓ_{1} : S^{1} → ℝ^{4} are continuous maps with lengths L_{i} = L(ℓ_{i}(S^{1})) for i = 1, 2, and with sup_{S^{1}} d(ℓ_{0}, ℓ_{1}) ≤ ε < L_{0} + L_{1}. There exists a (nonorientable) surface Σ with two boundary components C_{0}, C_{1} and a lagrangian map ℓ : Σ → ℝ^{4} with ℓ = ℓ_{i} on C_{i} for i = 1, 2, and A(ℓ(Σ)) ≤ c(L_{0} + L_{1})ε.

With this preparation we may now prove the following:

Proposition 7.8. For any (p, q) with (p, q) ≠ (1, 1), the cone C does not minimize area among nonorientable lagrangian comparison surfaces. In fact, there exists a nonorientable lagrangian surface Σ with ∂Σ = C ∩ ∂B_{1}, and A(Σ) < A(C ∩ B_{1}).

Proof. Because of the homogeneity of the cone, it clearly suffices to work in a ball of any radius. We let Λ > 1 be a large number to be determined, and observe that the mean curvature vector H of C is homogeneous of degree −1 and is therefore bounded on C ∩ (B_{Λ} \ B_{1}). We may then do a normal variation of C by an amount ε_{0}H for a fixed ε_{0} > 0 to produce a lagrangian perturbation C_{1} of C ∩ (B_{Λ} \ B_{1}) with A(C_{1}) ≤ A(C ∩ (B_{Λ} \ B_{1})) − c log(Λ) (since the integral of H^{2} on C ∩(B_{Λ} \B_{1}) is of order log(Λ)). Now we write ∂C_{1} = Γ_{1} ∪Γ_{2} where Γ_{1} is a curve of bounded distance to C ∩∂B_{1}, and Γ_{2} is of distance at most cΛ^{−1} to C ∩ ∂B_{Λ}. Applying Lemma 7.7 , we may find nonorientable lagrangian “strips” Σ_{1}, Σ_{2} of bounded area and with ∂Σ_{1} = Γ_{1} ∪ (C ∩ ∂B_{1}), ∂Σ_{2} = Γ_{2} ∪ (C ∩ ∂B_{Λ}). Thus we may form the nonorientable surface Σ which is equal to a union of Σ_{1}, C_{1}, Σ_{2}, and C ∩ B_{1}. We then have ∂Σ = C ∩ ∂B_{Λ}, and A(Σ) ≤ c + A(C_{1}) ≤ A(C ∩ B_{Λ}) + c − c log(Λ). If Λ is chosen sufficiently large this gives us a comparison surface which is smaller than the cone. q.e.d.

We next consider the minimizing properties (among orientable comparisons) of the non-multiply covered (p, p + 1) cones. Set,

γ(s) = \frac{1}{\sqrt{2p + 1}} (\sqrt{p + 1}e^{i\sqrt{\frac{p}{p+1}} s}, i\sqrt{p}e^{−i\sqrt{\frac{p+1}{p}} s}) ,

where 0 ≤ s ≤ 2π\sqrt{p(p + 1)}.

Theorem 7.9. For at least one integer p ≥ 1, the hamiltonian stationary lagrangian (p, p + 1) cone minimizes area among disk type lagrangian comparison surfaces with boundary γ(s).

Proof. Let α ∈ H_{2}(N, ℤ) be a lagrangian homology class with c_{1}(N)(α) ≠ 0. The existence theorem allows us to represent α by a finite set of weakly lagrangian maps that are area minimizers. For at least one such map ℓ : Σ → N we have c_{1}(N)(ℓ(Σ)) ≠ 0. The regularity theorem implies that ℓ is a branched immersion except at finitely many cone-type isolated singularities. By Propostion 2.2 twice the sum of the local Maslov indices at these singularities equals c_{1}(N)(ℓ(Σ)) and is therefore not zero. Hence there must be a nontrivial area minimizing cone. q.e.d.

Example. Perhaps the simplest example of a Kähler surface (N, ω) and a homology class α ∈ H_{2}(N, ℤ) satisfying [ω](α) = 0 and c_{1}(N)(α) ≠ 0 is given as follows: Let N be ℙ^{2} with one point blown up. Let L be the class of a line in ℙ^{2} and E be the class of the exceptional curve. There is a Kähler form ω on N satisfying [ω](L) = [ω](E) so that L − E is a lagrangian class. The canonical class is K = 3L − E and therefore c_{1}(N)(L − E) = 2. There are, of course, many such examples.

In fact it is possible to show that the (p, p + 1) cones occur in situations where c_{1}(N) = 0. To discuss these examples we digress to describe some results due to Hitchin [H] on Einstein four-manifolds. We are indebted to C. LeBrun for referring us to the relevent literature. Hitchin proves the following result: Let N be a compact four-dimensional Einstein manifold with signature σ and Euler characteristic χ. Then

|σ| ≤ \frac{2}{3}χ.

If equality occurs then ±N is either flat or its universal cover is a K3 surface. If the universal cover of N is a K3 surface then N is a K3 surface, an Enriques surface or the quotient of an Enriques surface by a free antiholomorphic involution. In a note added in proof, Hitchin shows that there is a K3 surface X, given as a complete intersection in ℂP^{5}, that admits a free holomorphic involution τ_{+} and a free antiholomorphic involution τ_{−} that commute and such that τ_{+}τ_{−} is free. Also both involutions are isometries of the metric h induced on X by the Fubini-Study metric.

Denote the Kähler form of h by ω_{h}. Since τ_{+} is a holomorphic isometry, τ_{+}^{∗}ω_{h} = ω_{h} and therefore ω_{h} descends to a Kähler form on the quotient four manifold Y_{+} = X/τ_{+}. It follows easily that Y_{+} is an Enriques surface. By Yau’s theorem [Y] there exists a unique Ricci-flat Kähler metric g_{+} on Y_{+} with Kähler form ω_{g_{+}} in the same cohomology class as ω_{h}. This metric lifts to a Ricci-flat Kähler metric on X that we denote by g. Note that τ_{+} is a holomorphic isometry of g. Next consider the antiholomorphic involution τ_{−}. Let σ denote a nonzero holomorphic (2, 0)-form on X. With respect to g, σ is a parallel section of the canonical bundle. Then τ_{−}^{∗}σ is a holomorphic (2, 0)-form for the conjugate complex structure and so is a parallel section of the anticanonical bundle. Thus τ_{−}^{∗}σ = ±σ. It follows that either Re σ or Im σ is invariant under τ_{−}. Suppose that Re σ is invariant under τ_{−}. (The case that Im σ is invariant under τ_{−} is entirely similar.) By scaling we can assume that Re σ has unit length. Then Re σ determines a complex structure J_{σ} that lies on the same hyperKähler line as the original complex structure and Re σ is the Kähler form for the metric g and J_{σ}. Since both Re σ and J_{σ} are invariant under τ_{−} so is g. Therefore both τ_{+} and τ_{−} are isometries of the Calabi-Yau metric g. It follows that there is a Ricci-flat Kähler metric on the Enriques surface Y_{+} for which τ_{−} is an antiholomorphic isometry and a Ricci-flat metric on Z = Y_{+}/τ_{−}. Note that π_{1}(Z) = ℤ_{2} × ℤ_{2} and b_{2}^{+}(Z) = 0. Thus Z cannot be Kähler or symplectic. The Kähler form ω_{g_{+}} on Y_{+} is taken to −ω_{g_{+}} by τ_{−} and so does not descend to Z. However the density |ω|, where ω = ω_{g_{+}}, is well-defined on Z and we can define the notion of a lagrangian in Z. We say a map ℓ : Σ → Z is lagrangian if ℓ^{∗}(|ω|) = 0. Note that the lift of a lagrangian to Y_{+} is lagrangian in the usual sense. Most of the existence and regularity results of this paper easily extend to this generalized notion of lagrangian.

Note that τ_{+} : X → X is the covering involution for the covering map π_{+} : X → Y_{+}. The canonical bundle K_{X} on X is trivial. Let ξ denote a parallel section of K_{X} of unit length. Then τ_{+}^{∗}(ξ) is also a parallel section of unit length. Since τ_{+} is a holomorphic involution τ_{+}^{∗}(ξ) = ±ξ. However if τ_{+}^{∗}(ξ) = ξ then ξ descends to a parallel section of K_{Y} the canonical bundle on Y. Thus K_{Y} is trivial. This contradiction implies that τ_{+}^{∗}(ξ) = −ξ.

Let f : T^{2} → Z be an incompressible immersion. The immersion f is double covered by an immersion f̃: T^{2} → Y_{+}. Clearly

∫_{T^{2}} f̃^{∗}ω = 0.

It follows that in a tubular neighborhood of f̃(T^{2}) we have ω = dη, for a one-form η. Note that τ_{−}^{∗}η = −η, so η does not descend to Z, though the density |η| is well-defined in a neighborhood of f(T^{2}). Suppose that f(T^{2}) has a fine triangulation such that a neighborhood of f(T^{2}) is covered by open Darboux balls with the properties that each closed 2-simplex on f(T^{2}) lies in at least one ball of the cover and such that if a 1-simplex intersects a ball in the cover then it lies entirely in the ball. As in the proof of Proposition 2.1 we can perturb each 1-simplex, λ, keeping its endpoints fixed, so that on the perturbed 1-simplex, λ′, we have ∫_{±λ′} η = 0. Then the integral of ±η around the boundary of each perturbed 2-simplex is zero and the 2-simplex can be replaced by a lagrangian simplex. The resulting piecewise C^{1} lagrangian map ℓ_{0} : T^{2} → Z is a perturbation of f and so is incompressible.

We next minimize area among lagrangian maps ℓ : T^{2} → Z with the same induced map on π_{1} as that of ℓ_{0}. By the existence and regularity theory developed in the previous sections a minimizer ℓ_{Z} exists. The map ℓ_{Z} : T^{2} → Z is a Lipschitz lagrangian map that is smooth except (perhaps) at finitely many points. These points are (p, p + 1) cone-type singularities described precisely above. The induced map ℓ_{Z}_{∗} on π_{1} is the same as that of ℓ_{0}. We suppose, by way of contradiction, that ℓ_{Z} is smooth everywhere. Then ℓ_{Z} is double covered by a smooth lagrangian map ℓ_{Y} : T^{2} → Y_{+} which is equivariant with respect to an involution on T^{2} and the antiholomorphic involution τ_{−}. Clearly ℓ_{Y} minimizes area among such lagrangian maps. In particular, since Y_{+} is Kähler-Einstein and ℓ_{Y} is smooth the mean curvature H is an admissible variation. Using H in the first variation formula it follows that ℓ_{Y} is a minimal (H = 0) branched immersion. Since one generator of π_{1}(T^{2}) is mapped to a nontrivial element of π_{1}(Y_{+}) = ℤ_{2}, ℓ_{Y} is double covered by a minimal lagrangian map ℓ_{X} : T^{2} → X. Denote the image of ℓ_{X} by Σ and note that τ_{+}(Σ) = Σ. Since Σ is minimal lagrangian it is calibrated by a parallel unit section ξ of K_{X}. Thus,

∫_{Σ} Re(ξ) = ∫_{Σ} dvol.

But τ_{+}^{∗}(ξ) = −ξ and therefore

∫_{Σ} Re(ξ) = − ∫_{Σ} dvol.

We conclude that ∫_{Σ} dvol = 0 and therefore the map ℓ_{X} is a map to a point. Thus the map ℓ_{Z} is also a map to a point. But this is impossible since it is incompressible. This contradiction implies that ℓ_{Z} cannot be regular everywhere and hence that there are singular points on ℓ_{Z}. Taking coverings this implies that in the K3 surface X there is a lagrangian stationary torus that contains singular points.

8. Main results

We can combine the existence Theorem 5.21 , the regularity Theorem 4.10 and the instability of the hamiltonian stationary lagrangian (p, q) cones for |p − q| > 1 to prove:

Theorem 8.1. Let (N, ω, J) be a compact symplectic 4-manifold with compatible metric g. Then the lagrangian homology is generated by classes that can be represented by lagrangian Lipschitz maps ℓ that are branched immersions except at finite number of singular points. The local Maslov index of each singular point is either 1 or −1. Twice the sum of the local Maslov indices equals the first Chern class of N paired with the homology class of the surface. The maps ℓ are minimizers of area constrained to lagrangian cycles. The mean curvature H of each surface satisfies a first order elliptic system of “Hodge-type”.

Lemma 8.2. A branched lagrangian immersion in a Kähler-Ein-stein surface is stationary if and only if it is lagrangian stationary.

Proof. Since the ambient manifold is Kähler-Einstein σ_{H} = H ω is closed. To prove the lemma, we need only show that the mean curvature generates a smooth deformation through lagrangian submanifolds. We first consider the immersion case and then extend the argument to include branch points. Let ℓ : Σ → N be a lagrangian immersion. We can extend ℓ to be an immersion L from a neighborhood O of the zero section in the normal bundle of Σ where we identify the zero section with Σ. The map L is then a local diffeomorphism, and we can pull back both g and J to O using L. It suffices to construct a lagrangian variation of the zero section whose initial derivative is H. To accomplish this, let Π : O → Σ denote the natural projection map, and let σ = −Π^{∗}(σ_{H}). There is a unique vector field V such that ω(H, X) = g(V, X), and since σ is closed, V is locally a hamiltonian vector field. Also V = H on Σ. We let F_{t} denote the flow defined by V near Σ. The transformations F_{t} are then symplectic, and L(F_{t}(Σ)) gives a lagrangian variation. If we assume that ℓ is lagrangian stationary, then the first variation of area is zero for this variation. The first variation of area is given by minus the L^{2} inner product of the mean curvature vector field with the variation vector field. Therefore, it is strictly negative unless H = 0. Thus ℓ is stationary. This proves the lemma for immersions.

Let p_{1}, …, p_{k} ∈ Σ be the branch points of ℓ. Choose r > 0 such that the discs , D_{r}(p_{i}), of radius r centered at p_{i} are disjoint. On D_{r}(p_{i}) there is a smooth function, β_{i}, with β_{i}(p_{i}) = 0 such that σ_{H} = dβ_{i}. Choose local coordinates {x_{1}, x_{2}} on D_{r}(p_{i}) centered at p_{i}. For ε < r, define a function (h_{i})_{ε} on D_{r}(p_{i}) by

(h_{i})_{ε}(x) = \begin{cases}0, & |x| < ε^{2} \\ \begin{matrix}log & \frac{|x|}{ε^{2}} \\ log & \frac{1}{ε} \end{matrix} , & ε^{2} ≤ |x| ≤ ε \\ 1, & |x| > ε. \end{cases}

Then a simple computation yields lim_{ε→0} ∫ |∇(h_{i})_{ε}|^{2}dμ = 0. Define σ_{ε} = d((h_{i})_{ε}β_{i}) on D_{r}(p_{i}) and σ_{ε} = σ_{H} otherwise. The closed 1-form σ_{ε} vanishes near the branch points. Set V_{ε} ω = σ_{ε}. Since ℓ is an immersion on Σ \ ∪_{i}D_{ε^{2}/2}(p_{i}) and V_{ε} vanishes on ∪_{i}D_{ε^{2}}(p_{i}), for each ε > 0 the above construction gives a lagrangian variation of ℓ that fixes ℓ near the branch points. Suppose that ℓ is lagrangian stationary. Then the first variation of area is zero for each such variation. Thus we have for each ε > 0,

(8.1) 0 = − ∫ ⟨V_{ε}, H⟩dμ

= − ∫_{Σ\∪_{i=1}^{k}D_{ε}(p_{i})} |H|^{2}dμ − ∑_{i}^{k} ∫_{D_{ε}(p_{i})}⟨V_{ε}, H⟩dμ.

For each i:

|∫_{D_{ε}(p_{i})}⟨V_{ε}, H⟩dμ| ≤ ∫_{D_{ε}(p_{i})}(h_{i})_{ε}|∇β_{i}|^{2}dμ
+ ∫_{D_{ε}(p_{i})} |β_{i}||⟨∇(h_{i})_{ε}, ∇β_{i}⟩|dμ
≤ ∫_{D_{ε}(p_{i})}(h_{i})_{ε}|∇β_{i}|^{2}dμ

+ (∫_{D_{ε}(p_{i})} |∇(h_{i})_{ε}|^{2}dμ)^{\frac{1}{2}} (∫_{D_{ε}(p_{i})} β_{i}^{2}|∇β_{i}|^{2}dμ)^{\frac{1}{2}} .

Thus ∫_{D_{ε}(p_{i})}⟨V_{ε}, H⟩dμ → 0 as ε → 0. It follows from (8.1) that 0 = ∫_{Σ} |H|^{2}dμ and therefore that ℓ is stationary. q.e.d.

Proposition 8.3. Let (N, ω, J) be a compact Kähler-Einstein surface. Suppose the maps ℓ in Theorem 8.1 are all branched immersions. Then the lagrangian homology is generated by classes that can be represented by branched lagrangian immersions that are classical minimal surfaces.

Proof. Let ℓ : Σ → N be a minimizer. Since the ambient manifold is Kähler-Einstein the 1-form σ_{H} on Σ is harmonic. Thus if Σ = S^{2}, σ_{H} = 0 and therefore H = 0. If genus(Σ) ≥ 1 then Σ is non-collapsible. If γ is a separating simple closed curve then [γ] is zero in homology and therefore [σ_{H}]([γ]) = 0. In the construction of ℓ the period condition is only used for classes that are represented by separating curves on Σ. It follows that variation by the mean curvature preserves the noncollapsibility condition and therefore H is an admissible variation. The Proposition then follows from the lemma. q.e.d.

Recall that a homotopy class α ∈ π_{2}(N) is called lagrangian if there is a weakly lagrangian map ℓ ∈ W^{1,2}(S^{2}, N) that represents α. Applying the existence and regularity results we can represent a lagrangian homotopy class α by a sum of Lipschitz lagrangian maps ℓ_{λ} : S^{2} → N satisfying the same properties as the maps in Theorem 8.1.

Suppose that α ∈ π_{2}(N) is a lagrangian homotopy class. We define the lagrangian area of α:

Larea(α) = inf{area(ℓ) : ℓ ∈ W_{L}^{1,2}(S^{2}, N) and ℓ represents α}.

If m is an integer greater than 1, we say a lagrangian homotopy class α is m-stable if:

Larea(mα) = m Larea(α).

Theorem 8.4. Suppose that α ∈ π_{2}(N) is a lagrangian homotopy class that can be represented by a lagrangian minimizer ℓ : S^{2} → N. If α is m-stable then ℓ is a branched immersion (there are no singular points).

Proof. Suppose not. Then ℓ contains at least two singular points, one with Maslov index +1, one with Maslov index −1. Consider an m-covering of S^{2}, ϕ : S^{2} → S^{2}, branched at the two singular points. The map ℓ ∘ ϕ is lagrangian, represents mα and has A(ℓ ∘ ϕ) = mA(ℓ) = m Larea(α). It is therefore an area minimizer. But ℓ∘ϕ has two singular points with multiply covered tangent cones and is therefore unstable by Proposition 7.5. q.e.d.

Corollary 8.5. Under the assumptions of the theorem if, in addition, N is a Kähler-Einstein surface then the minimizer ℓ is a branched minimal lagrangian immersion.

Appendix A. The geometric equations

Suppose that N is a 2n-manifold with symplectic form ω, almost complex structure J and compatible metric g.

Proposition A.1. Let ℓ : Σ → N be a lagrangian immersion. Denote by H the mean curvature vector field of Σ in V and by Ric, the Ricci 2-form of g. Then

d(H ω) = d ∗ τ + ℓ^{∗}Ric.

Here τ is a one form on Σ determined by the torsion of a connection on N. The connection is uniquely determined by g and ω.

Proof. Exactly as in the Kähler case, the almost complex structure J determines an orthogonal splitting of the complexified tangent space, T N ⊗ ℂ, into +i and −i eigenspaces denoted, T^{(1,0)}N and T^{(0,1)}N, respectively. Let

{e_{1}, e_{2}, f_{1}, f_{2}} (A.1)

be an oriented orthonormal frame adapted to Σ in the sense that:

(i) {e_{1}, e_{2}} is an oriented frame of the lagrangian plane T Σ.

(ii) {f_{1}, f_{2}} is an oriented frame of the lagrangian plane T Σ^{⊥}.

(iii) Je_{j} = f_{j} j = 1, 2.

The vectors {u_{j} = e_{j} − if_{j}} form a unitary frame of T^{(1,0)}N and the vectors {u_{j} = e_{j} + if_{j}} form a unitary frame of T^{(0,1)}N. Thus we have:

g(u_{j}, u_{k}) = 0, g(u_{j}, u_{k}) = 2δ_{jk}. (A.2)

Let {θ_{1}, θ_{2}, η_{1}, η_{2}} be the orthonormal coframe dual to the orthonormal frame (A.1). From (i) it follows that

{θ_{1}, θ_{2}} is an orthonormal coframe on L for the induced metric. (A.3)

From (ii) it follows that η_{1} = η_{2} = 0 on Σ. (A.4) From (iii) it follows that Jη_{j} = θ_{j}, j = 1, 2. (A.5) The 1-forms ω_{j} = θ_{j} + iη_{j}, j = 1, 2, (A.6) form a unitary coframe adapted to Σ. They are dual to the unitary frame {u_{1}, u_{2}}.

Let ∇ denote a metric compatible connection for g. Then

∇u_{j} = ∑_{k} ω_{jk} ⊗ u_{k} + ∑_{k} τ_{jk} ⊗ u_{k}, j = 1, 2. (A.7)

From (A.2) it follows that ω_{jk} + ω_{kj} = 0, τ_{jk} + τ_{kj} = 0. (A.8) Thus the unitary coframe (A.6) satisfies the structure equations:

dω_{j} = ∑_{k} ω_{jk} ∧ ω_{k} + ∑_{k} τ_{jk} ∧ ω_{k}. (A.9)

In particular (ω_{jk}) is the connection one-form and (τ_{jk}) is the torsion form with respect to the coframe {ω_{1}, ω_{2}}. Note that the only nonzero component of the torsion is, τ_{12} = −τ_{21}. The symplectic form,

ω = \frac{i}{2} ∑_{j} ω_{j} ∧ ω_{j}.

satisfies dω = 0. This is equivalent to the condition: (A.10) τ_{12} is a form of type (0, 1).

minimizing area among lagrangian surfaces 83

We remark that the condition (A.10) implies that the connection defined by (A.7) is unique. Thus the symplectic form and the metric, together, determine a unique connection with torsion τ_{12}.

The connection form (ω_{jk}) can be written ω_{jk} = θ_{jk} + iη_{jk} (A.11) where θ_{jk} = −θ_{kj} and η_{jk} = η_{kj}. The torsion form (τ_{jk}) can be written τ_{jk} = σ_{jk} + iρ_{jk} (A.12) where σ_{jk} = −σ_{kj} and ρ_{jk} = −ρ_{kj}. The structure equations (A.9) become:

(A.13) dθ_{j} = ∑_{k}(θ_{jk} + σ_{jk}) ∧ θ_{k} − ∑_{k}(η_{jk} − ρ_{jk}) ∧ η_{k},
(A.14) dη_{j} = ∑_{k}(η_{jk} + ρ_{jk}) ∧ θ_{k} + ∑_{k}(θ_{jk} − σ_{jk}) ∧ η_{k}.

Since η_{j} = 0, j = 1, 2, along Σ these equations imply that on Σ:

∑_{k}(η_{jk} + ρ_{jk}) ∧ θ_{k} = 0. (A.15)

Thus,

η_{jk} + ρ_{jk} = ∑_{l} h_{jkl}θ_{l} (A.16)

where h_{jkl} = h_{jlk}. The {h_{jkl}} are the components of the second fundamental form of Σ in N. The mean curvature vector is:

H = ∑_{k,j} h_{jkk}f_{j}. (A.17)

Then,

J(H) = − ∑_{k,j} h_{jkk}e_{j}.

Dualizing we have J(H)^{#} = − ∑ h_{jkk}θ_{j}. (A.18)

Let ∗ denote the Hodge star operator on Λ^{1}(Σ). This operator depends only on the conformal structure on Σ. From (A.16) we have

J(H)^{#} = 2 ∗ ρ_{12} − ∑_{k} η_{kk}. (A.19)

where ρ_{12} = Im τ_{12}. By (A.11), iη_{kk} = ω_{kk} and thus we have

H⌟ ω = ∗τ + i ∑_{k} ω_{kk}, (A.20)

where we have set τ = 2ρ_{12}. Taking the exterior derivative the result follows. q.e.d.

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86 r. schoen & j. wolfson

Stanford University, Stanford, CA 94305 Michigan State University, East Lansing, MI 48824