j. differential geometry 58 (2001) 233-261

A CONSTRUCTION OF NEW FAMILIES OF MINIMAL LAGRANGIAN SUBMANIFOLDS VIA

TORUS ACTIONS

EDWARD GOLDSTEIN

Abstract

In this paper we investigate connections between minimal Lagrangian submanifolds and holomorphic vector fields in Kähler manifolds. Our main result is: Let M^{2n} (n ≥ 2) be a Kähler-Einstein manifold with positive scalar curvature with an effective, structure-preserving action by an n-torus T^{n}. Then precisely one regular orbit L of the T^{n}-action is a minimal Lagrangian submanifold of M. Moreover there is an (n − 1)-torus T^{n−1} ⊂ T^{n} and a sequence of non-flat immersed minimal Lagrangian tori L_{k} in M such that all L_{k} are invariant under T^{n−1} and L_{k} locally converge to L (in particular the supremum of the sectional curvatures of L_{k} and the distance between L and L_{k} go to 0 as k ↦ ∞). This result is new even for M = ℂP^{n} for n ≥ 3.

1. Introduction

This paper constitutes a part of author’s program in trying to understand the geometry of minimal Lagrangian submanifolds in Kähler-Einstein manifolds. Kähler-Einstein manifolds are just Kähler manifolds whose Ricci form is proportional to their Kähler form. The existence theory of Kähler-Einstein metrics on complex manifolds was one of the major recent achievements in complex differential geometry. One should mention the work of Aubin [2] and Yau [18] for the case of negative scalar curvature, Yau’s resolution of the Calabi conjecture and the existence of Calabi-Yau metrics [18] and the work of Tian [17] on the existence of Kähler-Einstein metrics with positive scalar curvature.

On a given Kähler-Einstein manifold M one has a class of minimal Lagrangian submanifolds of M. Those are just Lagrangian submanifolds of M, which in addition are minimal submanifolds of M, i.e., their mean curvature vector field vanishes. Compact minimal Lagrangian submanifolds of M are just the critical points of a natural variational problem, namely minimal Lagrangian submanifolds of M are the critical points for the volume functional on the space of Lagrangian submanifolds in M (see [15]).

Received March 15, 2001, and in revised form August 28, 2001.

The geometry of minimal Lagrangian submanifolds in a given Kähler-Einstein manifold M and the structure of their moduli-space depends strongly on the sign of the scalar curvature s of M. If the scalar curvature s of M is zero then M is a Calabi-Yau manifold and minimal Lagrangian submanifolds of M are just the Special Lagrangian submanifolds of M (see [16]). The Special Lagrangian submanifolds are calibrated and they come in smooth, finite-dimensional families. Special Lagrangian submanifolds have recently attracted a lot of attention due to their role in the Strominger-Yau-Zaslow mirror symmetry conjecture [16].

If the scalar curvature s of M is negative then minimal Lagrangian submanifolds of M have interesting strict volume minimization properties which we hope to explore in a forthcoming paper. In this paper we will mostly be interested in the case when the scalar curvature of M is positive. Our aim is to understand certain global properties that minimal Lagrangian submanifolds possess and to construct some new examples of minimal Lagrangian submanifolds. The main tools in our investigation will be holomorphic vector fields in the vicinity of minimal Lagrangian submanifolds on M (Section 2) and holomorphic isometries of M (Sections 3 and 4). Our main result is a construction of new families of minimal Lagrangian submanifolds in toric Kähler-Einstein manifolds (Theorem 4.3.1 in Section 4). The paper is organized as follows:

In Section 2 we study some global connections between minimal Lagrangian submanifolds and holomorphic vector fields in their vicinity in a Kähler manifold M. In Section 2.1 we present some basic properties of minimal Lagrangian submanifolds and holomorphic vector fields used throughout the paper. In Section 2.2 we prove the following Theorem:

Theorem 1. Let N be a Kähler manifold, L be a compact oriented immersed minimal Lagrangian submanifold of N without boundary and V be a holomorphic vector field defined in a neighbourhood of L in N. Then if div(V) is the holomorphic divergence of V, we have

∫_{L} div(V) = 0.

We also give a simple application of this theorem to minimal Lagrangian submanifolds in ℂP^{n} (see Corollary 2.2.1). While this theorem is not directly relevant to the main result of the paper, we nevertheless include it for its own interest.

In Sections 3 and 4 we use structure-preserving torus actions on Kähler-Einstein manifolds with positive scalar curvature to construct invariant minimal Lagrangian submanifolds. Our strategy for doing this is as follows:

We begin with Section 3.1 reviewing the geometry of the total space of the canonical bundle K(M) of a Kähler-Einstein manifold M with positive scalar curvature. We will see that K(M) is naturally a Calabi-Yau manifold. Moreover there is a correspondence between minimal Lagrangian submanifolds of M and Special Lagrangian submanifolds of K(M), invariant under a certain radial vector field Y on K(M) (see Lemmas 3.1.1 and 3.1.2 of Section 3.1). Thus we recast the problem of constructing minimal Lagrangian submanifolds of M to a problem of constructing Special Lagrangian submanifolds of K(M), invariant under the Y -flow.

In Section 3.2 we suppose that we have a structure-preserving action by a k-dimensional torus T^{k} on M. We will be interested in finding the T -invariant minimal Lagrangian submanifolds of M. Now the T^{k}-action on M induces a T^{k}-action on K(M). We will see that there are canonical moment maps μ on M and μ′ on K(M) for these actions. Let Z ⊂ M be the zero set of μ and π : K(M) → M be the projection. Then the zero set of μ′ on K(M) is Z′ = π^{−1}(Z). Suppose that T acts freely on Z′′ = Z′ −Z (here we view Z ⊂ M and M is the zero section of K(M)). Then we have a symplectic reduction Q = Z′′/T. We will see that Q has a natural holomorphic volume form φ′ and a Kähler metric ω′ and Special Lagrangian submanifolds of (Q, ω′, φ′) lift to T -invariant Special Lagrangian (SLag) submanifolds of K(M). Also the vector field Y is tangent to Z′′ and it projects to a vector field Y′ on Q. Thus we reduced the problem of finding T -invariant minimal Lagrangian submanifolds of M to a problem of finding SLag submanifolds of Q, invariant under the flow of Y′.

In Section 3.3 we assume that k = n−1. In that case Q had complex dimension 2. Let X ⊂ Z′′ be the set of elements of Z′′ of unit length in K(M) and S = X/T ⊂ Q. Then S is a compact, 3-dimensional submanifold of Q. We will see that there is a non-vanishing vector field W on S such that there is a correspondence between Y′-invariant SLag submanifolds of Q and the trajectories of the W -flow on S.

Next we would like to develop a criterion to see that T^{n−1} acts freely on Z′′. We also would like to understand the periodic orbits of the vector field W on S (to construct immersed minimal Lagrangian tori on M). We can do it if we assume that M is a toric K-E manifold (see Section 4). In this case we can prove the following theorem, which is the main theorem of this paper:

Theorem 2. Let M^{2n} (n ≥ 2) be a Kähler-Einstein manifold with positive scalar curvature with an effective structure-preserving T^{n}- action. Then precisely one regular orbit L of the T^{n}-action is a minimal Lagrangian submanifold of M. Moreover there is an (n − 1)-torus T^{n−1} ⊂ T^{n} and a sequence of non-flat immersed T^{n−1}-invariant minimal Lagrangian tori L_{k} ⊂ M such that L_{k} locally converge to L (in particular the supremum of the sectional curvatures of L_{k} and the distance between L and L_{k} go to 0 as k → ∞).

Here by local convergence we mean the following: The distance between L_{k} and L goes to 0 as k → ∞. Also for any point l ∈ L we can choose a neighbourhood U_{l} of l in M such that for k large enough L_{k} ⋂ U_{l} is a finite union ⋃ L_{k}^{j} of submanifolds of the form L_{k}^{j} = exp(v_{k}^{j})(L ⋂ U_{l}), where v_{k}^{j} is a normal vector field to L on L ⋂ U_{l}. Moreover any subsequence v_{k}^{j} converges to 0 in the C^{∞} topology as k → ∞.

This result is new even for M = ℂP^{n} for n ≥ 3. For n = 2 examples of non-flat S^{1}-invariant immersed minimal Lagrangian tori in ℂP^{2} were constructed in [5] and in [10] using harmonic maps. After our paper was completed D. Joyce published a preprint [11], in which he in particular constructs Special Lagrangian cones in ℂ^{n+1}, invariant under a linear action of T^{n−1}. Those cones project to non-flat immersed minimal Lagrangian tori in ℂP^{n}, invariant under T^{n−1}.

Acknowledgments. This paper was a part of the author’s work towards his Ph.D. at MIT. He wants to thank his advisor, Tom Mrowka, for continuing support.

2. Global connections between holomorphic vector fields and minimal Lagrangian submanifolds

Let (M^{2n}, ω) be a Kähler manifold. In this section we will establish a global relation between minimal Lagrangian submanifolds of M and holomorphic vector fields in their vicinity on M (Theorem 2.2.1 of Section 2.2). We begin however with Section 2.1, presenting the basic facts needed throughout the paper:

2.1 Basic properties

Let (M^{2n}, ω) be a Kähler manifold. In this section we will discuss holomorphic vector fields on M and present some basic facts on minimal Lagrangian submanifolds of M. The results of this section are essentially known, though they are often stated in different terms in the literature.

First we discuss holomorphic vector fields on M. Let V be a vector field defined on some open subset U of M. The following proposition is elementary and well known (see [12], Proposition 4.1):

Proposition 2.1.1. The following conditions are equivalent:

1) The flow of V commutes with the complex structure J on M.

2) For any point m ∈ U the endomorphism X ↦ ∇_{X}V of T_{m}M is J-linear on T_{m}M.

3) The vector field V − iJV gives a holomorphic section of T^{(1,0)}U.

A vector field V satisfying the conditions of Proposition 2.1.1 is called a holomorphic vector field. Let V be a holomorphic vector field on some open subset U of M and let m be a point in U. Since the endomorphism X → ∇_{X}V is J-linear on T_{m}M we can define

(1) div(V) = trace_{ℂ}(X → ∇_{X}V).

Let f = Re f + iIm f be a holomorphic function on U. From Condition 3) of Proposition 2.1.1 we deduce that the vector field fV = Re fV + Im fJV is a holomorphic vector field on U. Moreover one easily computes that

div(fV) = fdiv(V) + V (f). (2)

Let K(M) be the canonical bundle of M, i.e., K(M) = Λ^{(n,0)}T^{∗}M, the bundle of (n, 0)-forms on M. Let φ be a section of K(M) over an open subset U of M (not necessarily a holomorphic section). Thus φ is an (n, 0)-form on U.

Proposition 2.1.2. Let V be a holomorphic vector field on U. Then

div(V)φ = L_{V} φ − ∇_{V} φ.

Proof. Let m ∈ U. Pick a unitary basis X_{1}, …, X_{n} of T_{m}M (here T_{m}M is viewed as Hermitian vector space with the complex structure J). Extend X_{i} to a unitary frame in a neighbourhood of m. Then

(3)  L_{V} φ(X_{1}, …, X_{n}) = V (φ(X_{1}, …, X_{n}))
− Σφ(X_{1}, …, [V, X_{n}], …, X_{n})

and

(4)  ∇_{V} φ(X_{1}, …, X_{n}) = V (φ(X_{1}, …, X_{n}))
− Σφ(X_{1}, …, ∇_{V} X_{i}, …, X_{n}).

Now ∇_{V} X_{i} = [V, X_{i}] + ∇_{X_{i}}V. We plug this into (4) and subtract (4) from (3) to deduce the statement of the proposition. q.e.d.

The following result is due to A. Futaki [6] and we include its proof for reader’s convenience:

Lemma 2.1.1. Let (M^{2n}, ω) be a Kähler-Einstein manifold with non-zero scalar curvature s and let t = s/2n. Let V be a holomorphic infinitesimal isometry defined on some neighbourhood U of M. Then the function μ = it^{−1}div(V) is a moment map for the V -action on (M, ω).

Proof. We need to prove that dμ = i_{V} ω. We shall prove it at a point m such that V (m) ≠ 0. Pick an element φ of K(M) over m which has unit length. Since the flow of V is given by holomorphic isometries we can extend φ to a unit length section of K(M) invariant under the V - flow on some neighbourhood U of m. The section φ defines a connection 1-form ξ on U, given by ξ ⊗ φ = ∇φ. The Einstein condition tells that

idξ = tω. (5)

Since φ is V -invariant we deduce from Proposition 2.1.2 that

div(V) = −ξ(V). (6)

Also since φ is V -invariant and the flow of V is given by isometries, we deduce that ξ is also V -invariant. Thus

0 = L_{V} ξ = d(ξ(V)) + i_{V} dξ
= −d(div(V)) − iti_{V} ω by (5) and (6)

and the lemma follows. q.e.d.

Next we discuss minimal Lagrangian submanifolds on M. Let L be an oriented Lagrangian submanifold of M. For any point l ∈ L there is a unique element κ_{l} of K(M) over l which restricts to the volume form on L (in the induced metric from M). Various k_{l} give rise to a section

κ_{L} : L → K(M). (7)

The section κ_{L} has constant length \sqrt{2^{n}} over L and it defines a connection 1-form ξ on L for the connection on K(M) over L by the condition ξ ⊗ κ_{L} = ∇κ_{L}. Here ∇ is the connection on K(M), induced from the Levi-Civita connection on M. Since κ_{L} has constant length, ξ is an imaginary valued 1-form on L.

Let h be the trace of the second fundamental form of L (the mean curvature vector field of L). So h is a section of the normal bundle of L in M and we have a corresponding 1-form σ = i_{h}ω on L. The following fact is well-known, although it is often stated a bit differently in the literature (see [13] and [15]):

Lemma 2.1.2. σ = iξ.

Proof. Let l ∈ L and e be some vector in the tangent space to L at l. To compute ξ(e) we need to compute ∇_{e}κ_{L}. Take an orthonormal frame (v_{j}) of T_{l}L and extend it to an orthonormal frame in a neighbourhood U of l in L such that ∇^{L}v_{i} = 0 at l (here ∇^{L} is the Levi-Civita connection of L). We get that

∇_{e}κ_{L} = κ_{L} · ∇_{e}κ_{L}(v_{1}, …, v_{n})
= κ_{L}(e(κ_{L}(v_{1}, …, v_{n})) − Σκ_{L}(v_{1}, …, ∇_{e}v_{j}, …, v_{n})).

Now e(κ_{L}(v_{1}, …, v_{n})) = 0. Also clearly

κ_{L}(v_{1}, …, ∇_{e}v_{j}, …, v_{n}) = i⟨∇_{e}v_{j}, Jv_{j}⟩
= i⟨∇_{v_{j}}e, Jv_{j}⟩ = i⟨−e, J(∇_{v_{j}}v_{j})⟩.

Here J is the complex structure on M. Thus we get that

∇_{e}κ_{L} = −i⟨e, Jh⟩κ_{L} = −iσ(e)κ_{l}.

Here h = Σ∇_{v_{j}}v_{j} is the trace of the second fundamental form of L. Thus σ = iξ. q.e.d.

Thus L is minimal (i.e., h = 0) iff κ_{L} is parallel over L.

2.2 Integral of the divergence

We now can state and prove the main theorem of Section 2:

Theorem 2.2.1. Let M be a Kähler manifold, L be a compact oriented immersed minimal Lagrangian submanifold of M without boundary and V be a holomorphic vector field defined in a neighbourhood of L in M. Then

∫_{L} div(V) = 0.

Proof. Let L be a minimal Lagrangian submanifold of M and V be a holomorphic vector field defined in a neighbourhood of L in M. Let κ_{L} be a section of K(M) over L as in Equation (7). Since κ_{L} restricts to the volume form on L we have ∫_{L} div(V) = ∫_{L} div(V)κ_{L}. Let

ϕ = i_{V} κ_{L}|_{L}. (8)

ϕ is an (n − 1)-form on L. We claim that

dϕ = div(V)κ_{L}|_{L}. (9)

Thus the assertion of the theorem will follow. To prove (9) let l be a point in L. By Lemma 2.1.2 we have that for any element w in the tangent bundle to L, ∇_{w}κ_{L} = 0. We can extend κ_{L} to a section κ′_{L} of K(M) over some neighbourhood Z of l in M such that for any element w in the normal bundle of L to M in Z ⋂ L we’ll have ∇_{w}κ′_{L} = 0. Thus we’ll have ∇κ′_{L} = 0 at every point in L ⋂ Z (here ∇ is the covariant derivative on M). This implies that dκ′_{L} = 0 at every point in L ⋂ Z (here again the exterior derivative d is on M). Now we use Proposition 2.1.2 for V and φ = κ′_{L}. We deduce that

div(V)κ′_{L} = L_{V} κ′_{L}

at every point in L ⋂ Z. Also L_{V} κ′_{L} = d(i_{V} κ_{L}) + i_{V} (dκ_{L}) and dκ′_{L} = 0 along L. Thus we get

div(V)κ_{L} = dϕ.

q.e.d.

Let us derive a simple corollary of Theorem 2.2.1:

Corollary 2.2.1. Let L be a compact immersed oriented minimal Lagrangian submanifold of ℂP^{n} and let (z_{1}, …, z_{n+1}) be the homogeneous coordinates on ℂP^{n}. Then we can’t have |z_{1}| > |z_{2}| at all points of L.

Proof. Consider the following circle action on ℂP^{n}:

e^{iθ}(z_{1}, …, z_{n+1}) = (e^{iθ}z_{1}, e^{−iθ}z_{2}, z_{3}, …, z_{n+1}).

Let V be the vector field on ℂP^{n} generating this action. ℂP^{n} is Kähler-Einstein with scalar curvature 2n, hence by Lemma 2.1.1 the function idiv(V) is a moment map for the S^{1}-action on ℂP^{n}. We have computed in [7] that

idiv(V) = (|z_{1}|^{2} − |z_{2}|^{2})/Σ|z_{i}|^{2}.

In fact we can also deduce this from Theorem 2.2.1. Indeed the map f = (|z_{1}|^{2} − |z_{2}|^{2})/Σ|z_{i}|^{2} is a moment map for the S^{1}-action on ℂP^{n}, hence it differs from idiv(V) by a constant c. Also the submanifold L′ = ((z_{1}, …, z_{n+1})||z_{1}| = |z_{j}|) is a minimal Lagrangian submanifold of ℂP^{n} (this is the Clifford torus, see [13]). Hence by Theorem 2.2.1 ∫_{L′} div(V) = 0. From this we deduce that c = 0, i.e., idiv(V) = f.

Let now L be an immersed oriented minimal Lagrangian submanifold of ℂP^{n}. We have ∫_{L} div(V) = 0. Hence we obviously can’t have |z_{1}| > |z_{2}| everywhere on L. q.e.d.

3. Minimal Lagrangian submanifolds and the symplectic reduction of the canonical bundle

3.1 The Calabi construction and the connection between minimal and Special Lagrangian submanifolds

Let (M^{2n}, ω) be a Kähler-Einstein (K-E) manifold with positive scalar curvature. In this section we will show that the total space K(M) of the canonical bundle of M has a natural Calabi-Yau structure. Moreover there is a correspondence between minimal Lagrangian submanifolds of M and certain Special Lagrangian submanifolds of K(M). We begin this section by studying the geometry of K(M).

Let K(M) be the total space of the canonical bundle of M^{2n} and π : K(M) → M be the projection. There is a canonical (n, 0)-form ρ on K(M) defined by ρ(a)(v_{1}, …, v_{n}) = a(π_{∗}(v_{1}), …, π_{∗}(v_{n})). Here a ∈ K(M) and v_{1}, …, v_{n} are tangent vectors to K(M) at a. The form φ = dρ is a holomorphic volume form on K(M). If z_{1}, …, z_{n} are local coordinates on M then dz_{1} ∧ … ∧ dz_{n} is a local section of K(M) over M which defines a coordinate function y on K(M). The collections of holomorphic functions (z_{1}, …, z_{n}, y) are coordinates on K(M) and

ρ = ydz_{1} ∧ … ∧ dz_{n} , φ = dy ∧ dz_{1} ∧ … ∧ dz_{n}. (10)

We also have a radial vector field Y on K(M), given at a point m ∈ K(M) by the vector m (viewed as a tangent vector to the linear fiber over π(m)). We have i_{Y} ρ = 0. Also the Lie derivative L_{Y} ρ = ρ. So

ρ = i_{Y} dρ = i_{Y} φ. (11)

For M a Kähler-Einstein manifold with positive scalar curvature E. Calabi has constructed a complete Ricci-flat Kähler metric on K(M) (see [4] and [14], p. 108). The metric is constructed as follows:

Definition 3.1.1. The connection on K(M) induces a horizontal distribution for the projection π, with the corresponding splitting of the tangent bundle of K(M) into horizontal and vertical distributions. We can identify the horizontal space at each point m ∈ K(M) with the tangent space to M at π(m). Let r^{2} : K(M) → ℝ_{+} be the square of the length of an element in K(M) and u : ℝ_{+} → ℝ_{+} be a positive smooth function with a positive first derivative. We define the metric ω_{u} on K(M) as follows: we put the horizontal and the vertical distributions to be orthogonal. On the horizontal distribution we define the metric to be u(r^{2})π^{∗}(ω) and on the vertical distribution we define it to be 2t^{−1}u′(r^{2})ω′. Here ω is the Kähler-Einstein metric on M, t is the scalar curvature of ω, divided by 2n (see Equation (5)) and ω′ is the induced metric on the linear fibers of π.

The Kähler-Einstein condition ensures that the corresponding 2form ω_{u} defining this metric on K(M) is closed, i.e., the metric is Kähler. If we take u(r^{2}) = (tr^{2} +l)^{\frac{1}{n+1}} , l > 0 then for the corresponding metric ω_{u} on K(M) we will have that the holomorphic (n + 1, 0)-form φ has constant length, hence φ is parallel and ω_{u} is Ricci flat. This metric is called the Calabi metric on K(M).

From now on we endow K(M) with a Kähler metric ω_{u} as above for any choice of the function u. We have a certain class of submanifolds of K(M), called the Special Lagrangian (SLag) submanifolds of K(M). We make the following general definition, taken from [8]:

Definition 3.1.2. Let (N^{2k}, ω) be a Kähler manifold of complex dimension k with a non-vanishing holomorphic (k, 0)-form φ. (N^{2k}, ω, φ) is called an almost Calabi-Yau manifold. Let P^{k} be an k-dimensional submanifold of N. Then P is called a Special Lagrangian (SLag) submanifold of N iff

ω|_{P} = 0 , Imφ|_{P} = 0.

We will not discuss the properties of SLag submanifolds here but refer the interested reader to [8] and [16].

We have an almost Calabi-Yau manifold (K(M), ω_{u}, φ) with the Kähler form ω_{u} and a holomorphic volume form φ constructed above. Let L be an oriented Lagrangian submanifold of M and let κ_{L} : L → K(M) be the canonical section (see Equation (7)). We define a submanifold L^{K} ⊂ K(M) by

L^{K} = (m ∈ K(M)|m = aκ_{L}(l) for l ∈ L , a ∈ ℝ). (12)

We have the following:

Lemma 3.1.1 ([7]). L is a minimal Lagrangian submanifold of M iff L^{K} is a Special Lagrangian submanifold of K(M)

Proof. First we note that L^{K} is Special, i.e., Imφ|_{L^{K}} = 0. Indeed Imρ|_{L^{K}} = 0 and hence Imφ|_{L^{K}} = d(Imρ|_{L^{K}}) = 0.

We now prove that L^{K} is Lagrangian iff L is minimal. Let m be a point on L^{K}, l = π(m) ∈ L and m = aκ_{L}(l) for a ∈ ℝ. The tangent space to L^{K} at m is spanned by κ_{L}(l) (viewed as a vertical vector in T_{m}K(M)) and by the vectors of the form (e + a∇_{e}κ_{L}). Here e is any tangent vector to L at l (viewed as an element of the horizontal distribution of T_{m}K(M)) and ∇_{e}κ_{L} lives in the vertical distribution of T_{m}K(M).

Let ξ be the connection 1-form on L defined by κ_{L} i.e., ∇κ_{L} = ξ⊗κ_{L}. Then ξ is a purely imaginary 1-form on L and ∇_{e}κ_{l} = ξ(e)κ_{L} for e ∈ T_{l}L. From this we easily deduce that L^{K} is a Lagrangian submanifold of K(M) iff ξ vanishes, which by Lemma 2.1.2 is equivalent to L being a minimal submanifold of M. q.e.d.

The manifold L^{K} is invariant under the flow of the vector field Y on K(M) (which is just scaling of K(M) by real numbers). Vice versa we have the following:

Lemma 3.1.2. Let L′ be a Special Lagrangian submanifold of K(M) −M, invariant under the flow of Y. Then L = π(L′) is an (immersed) minimal Lagrangian submanifold of M.

Proof. Let m ∈ L′. Since L′ is Lagrangian and Y is in the tangent space T_{m}L′ then the tangent space to L′ at m clearly decomposes as

T_{m}L′ = span(Y) ⊕ T′

and T′ is a subspace of the horizontal distribution to K(M) at m. The space π_{∗}(T′) can be viewed as the tangent space to L at l = π(m). Clearly this tangent space T_{l}L is Lagrangian, i.e., L is Lagrangian. Also L′ was Special and Equation (11) tells us that i_{Y} φ = ρ. Thus m (viewed as an (n, 0)-form on M at l) restricts to a real n-form on T_{l}L, i.e., m ∈ L^{K}. Hence locally L′ coincides with L^{K}. From Lemma 3.1.1 we deduce that L is minimal. q.e.d.

3.2 Torus actions and the symplectic reduction

In the previous section we showed how to find minimal Lagrangian submanifolds of M from certain SLag submanifolds of K(M). In this section we will see that if we have a structure-preserving torus action on M then we can find T -invariant SLag submanifolds of K(M) from SLag submanifolds of a certain symplectic reduction of K(M).

Let T^{k} act on M. Then this action induces a T^{k}-action on K(M). Let T be the Lie algebra of T, v ∈ T , X_{v} be its flow vector field on M and X′_{v} be its flow vector field on K(M). Let t > 0 be the scalar curvature of M, divided by 2n. Recall from Section 2, Lemma 2.1.1 that we have a natural moment map μ for the T^{k}-action on M given by

μ ⊗ v = −it^{−1}div(X_{v}). (13)

Here v ∈ T is an element of the Lie algebra of T, X_{v} if the flow vector field on M associated to v and ⊗ is the pairing between the Lie algebra and the dual Lie algebra of T. Let us now return to K(M). The T^{k}- action on M induces a T^{k}-action on K(M). We need to compute the moment map for the T^{k}-action on (K(M), ω_{u}) (see Definition 3.1.1). We have the following lemma, which we proved in [7]. Here we give a more direct proof, which we present for reader’s convenience:

Lemma 3.2.1. The map μ′ = −it^{−1}uπ^{−1}(σ) = uπ^{−1}(μ) is a moment map for the T^{k}-action on (K(M), ω_{u}).

Proof. Let v ∈ T be an element in the Lie algebra of T, X_{v} be the associated flow vector field on M and let X′_{v} be the associated flow vector field on K(M) for the T -action on K(M). Let m be a point on K(M).

It is enough to prove that for any tangent vector X in the tangent space T_{m}K(M) to K(M) at m we have dμ′(X) ⊗ v = ω_{u}(X′_{v}, X). Now the tangent space T_{m}K(M) naturally decomposes into the direct sum T_{m}K(M) = H ⊕ V of a horizontal and a vertical distributions for the connection on K(M) at m. By the construction of ω_{u} (Definition 3.1.1) we have that H and V are ω_{u}-orthogonal. We will prove that dμ′(X) ⊗ v = ω_{u}(X′_{v}, X) in two cases: X is in H or X is in V.

Case 1. X ∈ H. We have μ′ = u(r^{2})π^{−1}μ. Since X ∈ H we have X(r^{2}) = 0. Hence

dμ′(X) ⊗ v = u(r^{2})dμ(π_{∗}X) ⊗ v = u(r^{2})ω(X_{v}, π_{∗}X).

Also we have ω_{u}(X′_{v}, X) = u(r^{2})ω(X_{v}, π_{∗}X) and we are done.

Case 2. X ∈ V. In that case we have dπ^{−1}(μ)(X) = 0 and X(r^{2}) = 2⟨X, Y ⟩. Here Y ∈ V is the vector field as in (11) and ⟨ , ⟩ is the Riemannian metric on V. So

(14)  dμ′(X) ⊗ v = ⟨X, Y ⟩2u′(r^{2})μ ⊗ v
= −2iu′(r^{2})t^{−1}div(X_{v})ω′(Y, iX).

Here ω′ is the symplectic form on the linear fiber of K(M) through m as before.

To compute ω_{u}(X′_{v}, X) we need to understand what is the vertical component of X′_{v} at m. We claim that this component is equal to −div(X_{v})Y. It is enough to prove this claim in case X_{v} doesn’t vanish. Consider the trajectory m_{t} of m for the X′_{v}-flow on K(M). We can view m_{t} as a section of K(M) over the trajectory l_{t} of the X_{v}-flow on M through l = π(m). Moreover we have L_{X_{v}}m_{t} = 0. We also have that the vertical component of X′_{v} is equal to ∇_{X_{v}}m_{t}. But since L_{X_{v}}m_{t} = 0 we have by Proposition 2.1.2 of Section 2 that ∇_{X_{v}}m_{t} = −div(X_{v})m_{t} and the claim follows. So we have

(15)  ω_{u}(X′_{v}, X) = −2t^{−1}u′(r^{2})ω′(div(X_{v})Y, X)
= −2u′(r^{2})t^{−1}ω′(idiv(X_{v})Y, iX).

Since div(X_{v}) is purely imaginary, the right hand side of Equation (15) is equal to −2iu′(r^{2})t^{−1}div(X_{v})ω′(Y, iX). Comparing Equations (14) and (15) we get our claim. q.e.d.

Remark. We have seen in the proof of the previous lemma that if we pick an element v in the Lie Algebra of T^{k} then on K(M) − M the vector field X′_{v} is horizontal precisely along the zero set of μ′ ⊗ v. This fact will be useful later on.

Let now L′ be a (connected) SLag submanifold of K(M)−M, invariant under the T -action and under the Y -flow. Since L′ is Lagrangian and T -invariant, the moment map μ′ is constant on L′. But μ′ = uπ^{−1}(μ) and Y (μ′) = 2r^{2}u′π^{−1}(μ). So we must have π^{−1}(μ) = 0 on L′. Thus L′ must live on the zero set of the moment map μ′. Also from this argument and Lemma 3.1.1 we get that a T^{k}-invariant minimal Lagrangian submanifold L of M must live on the zero set of the moment map μ. Since those zero sets will be important in all subsequent discussions, let us assign them names:

Definition 3.2.1. Let Z ⊂ M be the zero set of the moment map μ on M and let Z′ = π^{−1}(Z) ⊂ K(M) be the zero set of the moment map μ′ on K(M). Let Z′′ = Z′ − Z ⊂ K(M). Let Q = Z′′/T. Let X be the intersection of the unit circle bundle of K(M) with Z′ and let S = X/T ⊂ Q.

We will illustrate the relation between various sets we introduced in a commutative diagram:

\begin{matrix}X ⊂ Z′′ ⊂ Z′ & ⟶^{ν′} & S ⊂ Q ⊂ Z′/T \\ ^{π}↓ & & _{π′}↓ \\ Z & ⟶^{ν} & M_{red} = Z/T. \end{matrix} (16)

From now on we assume that T acts freely on Z′′. This implies that Z′′ and Q are smooth. We will demonstrate examples where this holds in Section 4. We have a symplectic reduction M_{red} = Z/T and a (smooth) symplectic reduction Q = Z′′/T, endowed with a Kähler metric ω_{u}^{red} (see (16)). We will now see that Q has a natural non-vanishing holomorphic (n + 1 − k, 0)-form φ′, which a differential of a certain (n, 0)-form ρ′ on Q.

Let v_{1}, ..., v_{k} be a basis for the Lie algebra T of T^{k} and let X′_{1}, ..., X′_{k} be the corresponding flow vector fields on K(M). Let φ^{∗} = i_{X′_{1}} … i_{X′_{k}}φ be an (n − k + 1, 0)-form on K(M), obtained by contracting φ by X′_{1}, …, X′_{k}. Let ρ^{∗} = i_{X′_{1}} … i_{X′_{k}}ρ. We claim that

φ^{∗} = (−1)^{k}dρ^{∗}. (17)

We prove this by induction on k. Namely let φ_{l}^{∗} = i_{X′_{1}} … i_{X′_{l}}φ and ρ_{l}^{∗} = i_{X′_{1}} … i_{X′_{l}}ρ. We claim that φ_{l}^{∗} = (−1)^{l}dρ_{l}^{∗}. For l = 1 we have that ρ is X′_{1}-invariant. Hence

0 = L_{X′_{1}}ρ = dρ_{1}^{∗} + φ_{1}^{∗}.

Now we use induction. The form ρ_{l−1}^{∗} is X′_{l}-invariant. Hence

0 = L_{X′_{l}}ρ_{l−1}^{∗} = (−1)^{l−1}dρ_{l}^{∗} + φ_{l}^{∗}

and we are done by induction.

Both φ^{∗} and ρ^{∗} are T -invariant. Let ν : Z → M_{red} and ν′ : Z′′ → Q be the quotient maps (see (16)). One easily sees that there is a unique (n − k + 1, 0)-form φ′ on Q and a unique (n − k, 0)-form ρ′ on Q such that

(ν′)^{∗}(φ′) = φ^{∗} , (ν′)^{∗}(ρ′) = ρ^{∗} , φ′ = (−1)^{k}dρ′. (18)

We state our observations as a Proposition:

Proposition 3.2.1. (Q, ω_{u}^{red}, φ′) is an almost Calabi-Yau manifold as in Definition 3.1.2.

The vector field Y is tangent to Z′′ and T -invariant, hence it projects to a vector field Y′ on Q. We had i_{Y} φ = ρ on K(M) (see Equation (11)). Hence we have

i_{Y′}φ′ = (−1)^{k}ρ′ (19)

on Q. We obviously have the following:

Lemma 3.2.2. Let L′′ be a SLag submanifold of Q, invariant under the Y′-flow. Then L′ = (ν′)^{−1}(L′′) is a SLag submanifold of K(M), invariant under T^{k} and under the Y -flow. Here ν′ : Z′′ → Q is the quotient map.

3.3 Complexity one actions and periodic orbits

We continue to assume that we have a structure-preserving T^{k}-action on a compact Kähler-Einstein n-fold M with positive scalar curvature. In the previous section we have shown that one can reduce the problem of finding T^{k} and Y -invariant SLag submanifolds of K(M) to finding Y′-invariant SLag submanifolds of Q (provided that T^{k} acts freely on Z′′, see (16)). In this section we assume that k = n − 1. We continue to assume that T^{n−1} acts freely on Z′′. Let X ⊂ Z′′ and S = X/T ⊂ Q as in Definition 3.2.1. Thus S is a smooth compact 3-dimensional submanifold of a non-compact almost Calabi-Yau 2-fold Q. We will show that there is a vector field W on S such that there is a correspondence between Y′-invariant SLag submanifolds of Q and the trajectories of the W -flow on S.

As we saw the tangent bundle of K(M) decomposes as a direct sum V ⊕ H of the vertical and the horizontal distributions. Let U be the image of the Lie algebra of T under the differential of the action on K(M). At a point z ∈ Z′′, U(z) is an (n − 1)-dimensional vector subspace of T_{z}K(M), and it is contained in the horizontal distribution H at z (since on π(Z′′) the moment map μ vanishes, see Remark after the proof of Lemma 3.2.1). Also the Kähler form ω_{u} restricts to 0 on U. Let U^{c} be the complexification of U in the tangent bundle to K(M). Then U^{c} can be viewed as a complex (n − 1)-dimensional vector bundle over Z′′. Let H′ be the orthogonal complement of U^{c} in the horizontal distribution H along Z′′. Then the tangent bundle of K(M) along Z′′ splits a direct sum V ⊕ H′ ⊕ U^{c}. Also the quotient of V ⊕ H′ under the T -action can be identified with the tangent bundle to the symplectic reduction Q = Z′′/T. Since H′ and V are T -invariant the tangent bundle to Q splits as a direct sum of 2 complex line bundles:

T Q = V ⊕ H′. (20)

Also V and H′ are orthogonal both with respect to the symplectic form ω_{u}^{red} and the Riemannian metric on Q.

There is a natural circle action on X (see (16)), given by the multiplication by complex numbers of absolute value 1 on K(M). This action is T -invariant, hence it induces a circle action on S = X/T. Let F be the vector field generating this action on S. Then F = J(Y′) (here J is the complex structure on Q and Y′ is a vector field on Q as in Lemma 3.2.2). Also both Y′ and F are in the vertical distribution V along S and the tangent bundle T S of S splits as a direct sum

T S = H′ ⊕ span(F). (21)

Here H′ is the horizontal distribution of Q along S. Let γ be some path in S and let γ^{Q} be the orbit of γ under the Y′-flow in Q. We wish to understand when γ^{Q} is a SLag submanifold of Q. Let W be a tangent vector to γ. Clearly for γ^{Q} to be Lagrangian we need W to be ω_{u}^{red}-orthogonal to Y′, hence W must live in the horizontal distribution H′ along S. The form ρ′ = (−1)^{n−1}i_{Y′}φ′ (see (19)) is a (non-zero)

(1, 0)-form on H′. Hence the form Imρ′ has a 1-dimensional kernel in H′. Hence for γ^{Q} to be Special we need W to belong to the kernel of Imρ′. We can normalize W such that Reρ′(W) = 1. Those conditions give rise to a non-vanishing horizontal vector field W on S:

Definition 3.3.1. Let W be a vector field on S living in the horizontal distribution H′ along S (see (21)) such that Imρ′(W) = 0 and Reρ′(W) = 1.

Let γ be a trajectory of W on S and consider γ^{Q} ⊂ Q to be the image of γ under the Y′-flow. The forms ω_{u}^{red} and φ′ vanish on γ^{Q} along γ ⊂ γ^{Q}. Also the Y′-flow preserves the horizontal distribution and L_{Y′}ρ′ = ρ′. From this we easily deduce that γ^{Q} is a Y′-invariant SLag submanifold of Q. From Lemmas 3.1.2 and 3.2.2 we get the following:

Lemma 3.3.1. Let γ be a trajectory of W on S. Then L_{γ} = π((ν′)^{−1}(γ)) (see (16)) is an immersed minimal Lagrangian submanifold of M. If γ is periodic then L_{γ} is an immersed minimal Lagrangian torus.

There is one general relation among trajectories of W, which will later be important: Consider the circle action on K(M) as before. The (n, 0)-form ρ is equivariant with respect to this action, i.e., if λ ∈ S^{1} then λ^{∗}(ρ) = λρ. So ρ′ is also equivariant with respect to the circle action on Q. Also this action preserves the horizontal distribution H′ on S (see Equation (21)). Consider an element −1 ∈ S^{1}. Then −1^{∗}(ρ′) = −ρ′. From this we deduce that the −1-action on S reverses the vector field W (see Definition 3.3.1), i.e., −1_{∗}(W) = −W. We summarize this as:

Proposition 3.3.1. The −1-action on S sends W -trajectories to W -trajectories, but it reverses their directions.

4. Toric K-E manifolds

4.1 A generic subtorus

Let M be a Kähler-Einstein manifold with positive scalar curvature. In Section 3.3 we saw that if we have a T^{n−1}-action on M then one can construct minimal Lagrangian submanifolds of M from trajectories of the vector field W on S. In order to do this we needed T^{n−1} to act freely on Z′′ (see Definition 3.2.1). In this section we assume that M is toric, i.e., we have an effective structure-preserving T^{n}-action on M. For recent results on toric K-E manifolds we refer the reader to [17] and [3]. We will find a certain “generic” (n − 1)-subtorus T^{n−1} ⊂ T^{n} whose action on M accords with our requirements (in particular it will act freely on Z′′). We will use this subtorus T^{n−1} to construct immersed, T^{n−1}-invariant minimal Lagrangian submanifolds. But first we note that there is a unique minimal Lagrangian torus, invariant under the whole of T^{n}.

Suppose L is a regular orbit of the T^{n}-action on M (i.e., an orbit with a finite stabilizer). Then the canonical moment map μ for the T^{n}-action on M (see (13)) is constant on L, hence L is a Lagrangian submanifold of M. Suppose that L is a minimal submanifold of M. As we have seen in Section 3.2 we must have μ = 0 on L i.e., L ⊂ μ^{−1}(0). By Atyiah’s result [1], μ^{−1}(0) is connected, hence L = μ^{−1}(0). So if a regular orbit, which is a minimal submanifold of M exists, it must coincide with μ^{−1}(0). We note that such an orbit does exist:

Lemma 4.1.1 ([7]). There exists (a unique) regular orbit L of the T^{n}-action on M, which a is a minimal Lagrangian submanifold of M. L is precisely the zero set of the canonical moment map μ for the T^{n}- action on M as in (13).

Our next goal is to find a certain (n − 1)-dimensional subtorus T^{n−1} ⊂ T^{n} to proceed with the constructions in Section 3.3. This is done in the following lemma:

Lemma 4.1.2. Let M^{2n} be a K-E manifold with an effective T^{n}- action as above. Then there is an (n − 1)- torus T^{n−1} ⊂ T^{n} such that

i) The differential of the T^{n−1}-action on M is injective along Z and T^{n} acts freely on Z′′.

ii) There is an element v in the Lie algebra of T^{n−1} such that the flow vector field X_{v} doesn’t have a constant length along Z.

Here Z is the zero set of the canonical moment map for the T^{n−1} action on M as in (13) and Z′′ is given as in Definition 3.2.1.

Remark. Condition ii) in the lemma will be later used to show that certain minimal Lagrangian tori we shall construct have Killing fields of non-constant length, hence they are not flat.

Proof. Let T^{n−1} ⊂ T^{n} be some (n − 1)-torus. First we prove that if the differential of the T^{n−1}-action on M is injective along Z (the zero set of the canonical moment map of T^{n−1}), then the T^{n}-action on Z′′ is free. Suppose not. Then there is a point l ∈ Z′′ and an element 1 ≠ t ∈ T^{n} s.t. t · l = l. In that case t also preserves the points on the T^{n−1}-orbit through l. The tangent space P to this orbit at l is in the horizontal distribution at l (since we are on the zero set of the moment map of T^{n−1}). Also ω_{u}|_{P} = 0. So the differential dt of the t-action at l acts trivially on the complexification P^{c} of P in the tangent space T_{l}K(M). Also dt acts trivially on the vertical distribution V (l) at l. The vector space P^{c} ⊕ V (l) is a complex vector space of dimension n and dt acts trivially on it. Also dt preserves the holomorphic volume form φ on K(M) at l. Hence dt is trivial at l. Hence t acts trivially on K(M) and on M, but the T^{n}-action on M was effective- a contradiction.

Next we wish to understand for which (n − 1)-tori T^{n−1} ⊂ T^{n} the differential of the T^{n−1}-action is injective along the zero set Z of the canonical moment map of T^{n−1}. Let T^{∗} be the dual Lie algebra of T^{n} and let Λ ⊂ T^{∗} be the weight lattice of T^{n}. Any element 0 ≠ v ∈ Λ defines an (n−1)-torus T_{v} ⊂ T^{n} such that v vanishes on the Lie algebra of T_{v}. Let μ be the canonical moment map of T^{n} and μ_{v} be the canonical moment map of T_{v}. Then μ_{v} is just the restriction of μ to the dual Lie algebra of T^{n−1} = T_{v}. It is therefore clear that μ_{v} vanishes at a point l ∈ M iff μ(l) is proportional to v. Since M is a toric variety, the moment polytope is convex and has no faces in the interior. Since 0 is in the interior of the moment polytope by Lemma 4.1.1, it is clear that Z = μ^{−1}[t_{1}v, t_{2}v] with t_{1} < 0 < t_{2}. For any t_{1} < t < t_{2} the value tv is in the interior of the moment polytope, while t_{1}v and t_{2}v are not.

Let 0 ≠ v ∈ Λ and suppose the line span(v) ⊂ T^{∗} doesn’t intersect any of the (n − 2)-faces of the moment polytope of μ. This means that any point in Z has either a trivial or a 1-dimensional stabilizer in T^{n}. We claim that in this case the differential of the T^{n−1} = T_{v}-action is injective along Z. Suppose not. Then there is a point l ∈ Z and a vector 0 ≠ w in the Lie algebra of T^{n−1} such that the flow vector field X_{w} vanishes at l. Since l ∈ Z the flow vector field X′_{w} of w on K(M) is horizontal along π^{−1}(l) ⊂ K(M) (see Remark after the proof of Lemma 3.2.1). Hence X′_{w} vanishes along π^{−1}(l). Let g = exp(tw) for some t ∈ ℝ. Then the g-action on π^{−1}(l) is trivial. But this means that the differential dg of the g-action on the tangent space T_{l}M has Jacobian 1. Also g acts trivially on the orbit L′ of the T^{n}-action through l. The tangent space T_{l}L′ of L′ at l is (n − 1)-dimensional and ω restricts to 0 on it. Hence its complexification (T_{l}L′)^{c} in T_{l}M is a complex (n − 1)- dimensional space and dg acts trivially on it. Also dg has Jacobian 1.

Hence dg is trivial, hence g acts trivially- a contradiction.

A generic line in the projective space P T^{∗} doesn’t intersect the (n − 2)-faces of the moment polytope of μ. Also the set of lines passing through points of the weight lattice Λ is dense in P T^{∗}. So we can find v ∈ Λ so that Condition i) in our lemma holds for T^{n−1} = T_{v}. In order to ensure that ii) holds, consider a point b in the (n − 2)-face of the moment polytope (here we use the fact that n ≥ 2). The T^{n}-orbit μ^{−1}(b) has a stabilizer of dimension at least 2. Hence we can find a vector 0 ≠ w ∈ T in the Lie algebra of T^{n} such that b ⊗ w = 0 and the flow vector field X_{w} vanishes along μ^{−1}(b). We can find a sequence of elements v_{k} ∈ Λ such that the lines (v_{k}) = span(v_{k}) do not intersect the (n − 2)-faces of the moment polytope and (v_{k}) converge to the line (b) = span(b) in P T^{∗}. We can also find a sequence of vectors w_{k} ∈ T such that v_{k} ⊗ w_{k} = 0 and w_{k} converge to w.

Each v_{k} defines an (n − 1)-torus T_{k} ⊂ T. By our construction the vectors w_{k} are in the Lie algebra of T_{k}. Let μ_{k} be the canonical moment map of T_{k}, and Z_{k} be the zero set of μ_{k}. We can find points n_{k} on Z_{k} such that n_{k} converge to a point n ∈ μ^{−1}(b). Let X_{k} be the flow vector field of w_{k}. Then the length of X_{k} at points n_{k} goes to 0 as k → ∞. On the other hand the torus L = μ^{−1}(0) coming from Lemma 4.1.1 is contained in all of Z_{k}. Moreover the lengths of X_{k} along L are a-priori bounded from below. So we deduce that for k large enough the torus T^{n−1} = T_{k} satisfies Conditions i) and ii) of the lemma. q.e.d.

4.2 Flow-invariant function and its level sets

We continue to assume that we have a toric Kähler-Einstein n-fold M. From now on we pick a sub-torus T^{n−1} = T_{v} ⊂ T satisfying the conditions of Lemma 4.1.2. Here v is a non-zero element in the dual Lie algebra of T^{n} and v vanishes on the Lie algebra of T^{n−1}. To avoid confusion we explicitly note that the subsets X, Z, Z′′ and their reductions S, M_{red}, Q we’ve introduced in Equation (16) are for the T^{n−1}-action on M and K(M). From Lemma 3.3.1 we deduce that one can construct minimal Lagrangian submanifolds of M from the trajectories of the vector field W on S. Moreover periodic orbits of W give rise to immersed minimal Lagrangian tori in M. Our goal therefor is to understand the periodic orbits of W on the compact 3-manifold S. In this section we will show that there is a function f on S which is constant on the orbits of the W -flow on S. We will also investigate the salient features of the level sets of f. We construct the function f as follows:

The circle R = T/T^{n−1} acts freely on Q and on S by Lemma 4.1.2.

Definition 4.2.1. Let w ≠ 0 be some non-zero element in the Lie algebra of R. Let A_{w} be the flow vector field for the w-action on Q and on S.

The vector fields A_{w} and W commute (since the A_{w}-flow is structurepreserving and in particular in preserves the defining conditions 3.3.1 for W). We also have a (1, 0)-form ρ′ and a holomorphic (2, 0)-form φ′ on Q with φ′ = (−1)^{(n−1)}dρ′ (see Equation (18)). The flow of A_{w} preserves ρ′ and φ′. A key point in finding periodic trajectories of W is the fact that there is a function on S constant along the trajectories of W:

Lemma 4.2.1. Let h = ρ′(A_{w}) and f = Re(h). Then h is an S^{1}- equivariant function on S and f is constant along the trajectories of W on S.

Proof. The fact that h is S^{1}-equivariant on S follows from the fact that A_{w} is S^{1}-invariant (since the S^{1} and the T^{n}-actions on K(M) commute) and ρ′ is an S^{1}-equivariant (1, 0)-form on Q (see the discussion in the end of Section 3.3).

Next we prove that f is constant along the trajectories of W on S. We have:

0 = L_{A_{w}}ρ′ = d(i_{A_{w}}ρ′) + i_{A_{w}}dρ′ = dh + (−1)^{(n−1)}i_{A_{w}}φ′

(see Equation (18)). So dh = (−1)^{n}i_{A_{w}}φ′. So dh(W) = (−1)^{n}φ′(A_{w}, W). The vector field A_{w} is in the tangent bundle to S, hence we can decompose it into A_{w} = A_{w}^{H} + λF. Here A_{w}^{H} is the horizontal part of A_{w} (i.e., the part in the distribution H′; see Equation (21)), F is the generator of the S^{1}-action on S and λ ∈ ℝ. W is horizontal and H′ is a 1-dimensional complex vector bundle. The form φ′ is a (2, 0)-form on Q. Hence φ′(A_{w}^{H}, W) = 0. Also F = JY′. Hence φ′(F, W) = iφ′(Y′, W). By the construction of W we had that φ′(Y′, W) is real. From all this we deduce that dh(W) is purely imaginary, hence df(W) = 0. q.e.d.

From the previous lemma we deduce that the trajectories of W live on level sets of the function f. Next we need to understand those level sets in more detail.

We had our symplectic reductions M_{red} = Z/T^{n−1} and Q and we have a natural projection π′ : Q → M_{red} (see (16)). Let v be an element of the weight lattice Λ of T^{∗} defining the torus T^{n−1} = T_{v} (so v vanishes on the Lie algebra of T^{n−1}). As we have seen in the proof of Lemma 4.1.2, Z is equal to μ^{−1}[t_{1}v, t_{2}v] for t_{1} < 0 < t_{2}. Here μ is the canonical moment map for the T^{n}-action on M. We have the following:

Proposition-Definition 4.2.1. Let Z_{0} = μ^{−1}(t_{1}v, t_{2}v) and Z′_{0} = π^{−1}(Z_{0}) ⊂ K(M). T^{n−1} acts freely on Z_{0} and we have M_{0} = Z_{0}/T^{n−1} ⊂ M_{red}, which is the smooth part of M_{red}. Let S_{0} = (π′)^{−1}(M_{0}) ⋂ S = (Z′_{0} ⋂ X)/T^{n−1} (see (16)). Then the circle S^{1} acts freely on S_{0} and S_{0} is an S^{1} fiber bundle over M_{0}.

Let K_{i} = π^{−1}(μ^{−1}(t_{i}v)) ⋂ X, i = 1, 2 (here μ is the canonical moment map for the T^{n}-action on M). Let a_{i} = μ^{−1}(t_{i}v)/T^{n−1} ∈ M_{red} and K_{i} = K_{i}/T^{n−1} = (π′)^{−1}(a_{i}) ⋂ S (see (16)). Then each a_{i} is a point in M_{red}, K_{i} is a circle in S and S^{1} acts locally freely on K_{i}.

Proof. We have that T^{n} acts freely on M_{0}, and so in particular T^{n−1} acts freely on M_{0} and so S^{1} acts freely on S_{0}, i.e., S_{0} is an S^{1} fiber bundle over M_{0}.

Also μ^{−1}(t_{i}v) is connected (being a T^{n} orbit on M) and is (n − 1)- dimensional and T^{n−1} acts locally freely on it (see Lemma 4.1.2). From this it is clear that a_{i} = μ^{−1}(t_{i}v)/T^{n−1} ∈ M_{red} is 1 point. Also the T^{n−1} action is free on K_{i} and K_{i} is n-dimensional and connected, hence K_{i} is a circle in S. Also since the T^{n−1} action on μ^{−1}(t_{i}v) is locally free we deduce that the generating vector field F of the S^{1}-action on K_{i} (see (21)) is not contained in the image of the differential of the T^{n−1}-action on K_{i}, hence the S^{1}-action on K_{i} = K_{i}/T^{n−1} is locally free. q.e.d.

We will represent the setup in the previous Proposition-Definition in a commutative diagram.

\begin{matrix}Z′′ ⋂ X = (K_{i}) ⋃(Z′_{0} ⋂ X) & ⟶_{ν′} & S = (K_{i}) ⋃ S_{0} \\ ↓^{π} & & ↓_{π′} \\ Z = (μ^{−1}(t_{i}v)) ⋃ Z_{0} & ⟶_{ν} & M_{red} = (a_{i}) ⋃ M_{0}. \end{matrix} (22)

On Z_{0} (see Proposition-Definition 4.2.1) we have an oriented Lagrangian distribution D, given by the image of the Lie algebra T of T^{n} under the differential of the T^{n}-action on M. This distribution gives rise to a section κ of K(M) over Z_{0} such that κ restricts to the Riemannian volume form on this Lagrangian distribution. This section is T^{n}-invariant and has constant length \sqrt{2}^{n}. Hence it gives rise to an R = T^{n}/T^{n−1}- invariant section

κ′ = κ/\sqrt{2}^{n} : M_{0} → S_{0}. (23)

By definition κ′ restricts to a positive real n-form on the Lagrangian distribution D. From this we deduce that h = ρ′(A_{w}) is real and positive along κ′(M_{0}).

We can normalize w (the generator of the Lie algebra of R = T^{n}/T^{n−1}) such that v ⊗ w = 1 (here v as before is the element of the dual Lie algebra of T^{n} defining the sub-torus T^{n−1}). On M_{red} we have a function

τ = μ ⊗ w. (24)

The image of τ is the interval [t_{1}, t_{2}]. For each t_{1} ≤ t ≤ t_{2} the level set τ^{−1}(t) is an orbit of the R = T^{n}/T^{n−1}-action on M_{red}. We define:

Definition 4.2.2. Let L′ = τ^{−1}(0), L_{+} = κ′(L′) ⊂ S (see Equation (23)) and L_{−} = (−1) · L_{+}.

Each L_{±} is an orbit of the R-action on S. Also at points of L_{±} the vector field A_{w} is horizontal (since μ ⊗ w = 0 on L′) and ρ′(A_{w}) is real. The vector field W also satisfies those properties (see Definition 3.3.1), hence W is proportional to A_{w} along L_{±}. So we see that L_{±} are trajectories of W (of course the minimal Lagrangian torus of M coming from these trajectories as in Lemma 3.3.1 is the T^{n}-invariant minimal Lagrangian torus L = μ^{−1}(0)). We have the following:

Lemma 4.2.2. The differential df of f is non-vanishing on S − (L_{−} ⋃ L_{+}).

Proof. We have seen in the proof of Lemma 4.2.1 that

dh = (−1)^{n}i_{A_{w}}φ′.

On S′ = S ⋂(π′)^{−1}(M_{red} − L′) the vertical part of the vector field A_{w} doesn’t vanish (see the Remark after the proof of Lemma 3.2.1). Hence the form i_{A_{w}}φ′ restricts as a non-vanishing (1, 0)-form on the horizontal distribution H′ along S′ (see Equation (21)). From this it is clear that df|_{H′} ≠ 0 on S′.

On S ⋂(π′)^{−1}(L′) − (L_{−} ⋃ L_{+}) the function h is not real. Also h is equivariant with respect to the S^{1}-action. Let F be the vector field generating the S^{1}-action as before. Then the derivative of f = Reh is non-zero in the direction of F. q.e.d.

f attains a constant value f_{+} along L_{+} and a value f_{−} = −f_{+} along L_{−} = (−1) · L_{+}. Since S is compact and connected, it is clear from Lemma 4.2.2 that f_{+} is the absolute maximum of f, attained only at L_{+}, and f_{−} is the absolute minimum of f, attained only at L_{−}. Also for any s ∈ (f_{−}, f_{+}), the level set Σ_{s} = f^{−1}(s) is smooth. We will also need the following fact:

Proposition 4.2.1. f|_{K_{i}} = 0 (see Proposition-Definition 4.2.1).

Proof. Proposition-Definition 4.2.1 tells that the circles K_{i} are orbits for the S^{1}-action on S, hence the tangent space to K_{i} lives in the vertical distribution along S (see (21)). Also the R = T^{n}/T^{n−1}-action preserves K_{i}, so the vector field A_{w} is tangent to K_{i}, so A_{w} is vertical along K_{i}. But from this we deduce that h = ρ′(A_{w}) = 0 at K_{i}, and so f = 0 at K_{i}. q.e.d.

4.3 Periodic trajectories and the main theorem

In the previous section we saw that the orbits of the vector field W on S live on the level sets of the function f on S. Moreover all these level sets except L_{±} = f^{−1}(f_{±}) are smooth 2-dimensional submanifolds of S. We make the following definition:

Definition 4.3.1. Let Φ = f^{−1}(f_{−}, f_{+}).

Take any point m ∈ Φ and consider the level set Σ_{s} of f passing through m. From Lemma 4.2.2 we know that Σ_{s} is smooth. Let Σ_{s}^{0} be the connected component of Σ_{s} containing m. The vector field W is tangent to Σ_{s}^{0}. We have a free R = T^{n}/T^{n−1}-action on Σ_{s}^{0}, and this action preserves the vector field W.

Proposition 4.3.1. The vector field A_{w} (see Definition 4.2.1) is transversal to W at all points of Σ_{s}^{0}.

Proof. Let m′ ∈ Σ_{s}^{0}. If m′ ∈ S ⋂(π′)^{−1}(M_{red} − L′) (see Definition 4.2.2), then the vector field A_{w} is not horizontal at m′, so it can’t be proportional to W. If m′ ∈ S ⋂(π′)^{−1}(L′)−(L_{−} ⋃ L_{+}) then h = ρ′(A_{w}) is not real, while ρ′(W) is real. So again A_{w} and W can’t be proportional. q.e.d.

From Proposition 4.3.1 we get that the quotient of Σ_{s}^{0} by the Raction is a circle and W projects to a non-vanishing vector field on it. From this we deduce that the W -trajectory on S starting at m will intersect the R-orbit of m. Suppose it intersects this orbit for the first time at a point ξ(m)m, ξ(m) ∈ R. This gives rise to a well-defined function

ξ : Φ → R = T^{n}/T^{n−1}. (25)

Clearly ξ is continuous, R-invariant and constant along the trajectories of W. Also we have seen in Proposition 3.3.1 that the −1-action on S sends W -trajectories to W -trajectories in the reverse direction. From this we easily deduce that for any point m ∈ Φ:

ξ(−1 · m) = ξ(m)^{−1}. (26)

Obviously the trajectory through m is periodic iff ξ(m) is a root of unity in R. Let R′ be the set of roots of unity in R. Since ξ is continuous, ξ^{−1}(R′) will be everywhere dense in Φ unless ξ assumes a constant value not in R′ on some open subset of Φ. The next lemma shows that it is impossible:

Lemma 4.3.1. Suppose that ξ is constant on some open set U ⊂ Φ. Then ξ is equal to a constant g on the whole of Φ and g^{2} = 1.

Proof. Let S_{+} = f^{−1}(0, f_{+}), S_{−} = f^{−1}(f_{−}, 0). Thus S_{−} = −1 · S_{+}. Suppose that ξ is constant on some open set U ∈ Φ. Then ξ is constant on some open ball U′ either in S_{+} or in S_{−}. We can assume w.l.o.g. that U′ ⊂ S_{+}. We note that S_{+} is connected. In fact S_{+} is given by

S_{+} = (κ′(a)e^{iθ}|a ∈ M_{0} , − π/2 < θ < π/2) − L_{+}.

Here κ′ is given by Equation (23) and M_{0} is the smooth part of the symplectic reduction M_{red} (see (22)). First we prove that ξ is a constant g on S_{+}. Let A_{w}^{H} be the horizontal part of the vector field A_{w} (see Definition 4.2.1). Since S_{+} ⊂ π^{−1}(M_{0}) we deduce that A_{w}^{H} doesn’t vanish on S_{+}. We also note that on S_{+} the vector field A_{w}^{H} cannot be proportional to JW. Indeed suppose that A_{w}^{H} = λJW for some λ ∈ ℝ at some point m ∈ S_{+}. Then

h(m) = ρ′(A_{w}) = ρ′(A_{w}^{H}) = iλρ′(W).

So h(m) is purely imaginary, hence f(m) = 0 - a contradiction. Since both A_{w}^{H} and W lie in H′ (see Equation (21)), which is a complex 1dimensional distribution, we deduce that we can find a function b : S_{+} → ℝ such that on S_{+} the vector field W is pointwise proportional to a vector field

W′ = A_{w}^{H} + bJA_{w}^{H}. (27)

Hence the trajectories of W′ and W coincide. We also note that the W′-trajectories live on level sets of f which are compact submanifolds of S_{+}. Hence the W′-flow is complete on S_{+}. We will use W′ instead of W to prove that ξ is constant on S_{+}.

Definition 4.3.2. For a point m ∈ S_{+} let t(m) be the time for the W′-flow to hit the R-orbit of m for the first time.

We have the following:

Lemma 4.3.2. For any point m ∈ S_{+} we have

ξ(m) = exp(t(m)w).

Proof. Let γ be the trajectory of W′ through m and γ′ = π′(γ) be the corresponding path in M_{0} (here π′ : S_{+} → M_{0} is the natural projection; see (22)). We have a free R-action on M_{0} and the corresponding flow vector field B_{w} for the w-flow on M_{0} (here w is the generator of the Lie algebra of R). We obviously have π′_{∗}(A_{w}^{H}) = B_{w}. Hence the tangent field to γ′ is B_{w} + bJB_{w}.

The R-action on M_{0} is Hamiltonian with the moment map τ = μ ⊗ w (see (24)). Also the B_{w}-flow on M_{0} commutes with the complex structure J on M_{0}. Hence the vector fields B_{w} and JB_{w} commute. Let P_{1} = γ′(0) and P_{2} = γ′(t(m)). Then P_{2} = ξ(m)P_{1}. We will prove that P_{2} = exp(t(m)w)P_{1} and since the R-action on M_{0} is free this would prove that exp(t(m)w) = ξ(m).

Let exp(xJB_{w}) be the time x flow of JB_{w}. Note that the JB_{w}-flow is not complete. In fact we have

JB_{w}(τ) = ω_{red}(B_{w}, JB_{w}) = |B_{w}|^{2} > 0.

So τ increases on the JB_{w}-trajectories. Let

c(r) = ∫_{[0,r]} b(γ(t))dt for 0 ≤ r ≤ t(m). (28)

Consider a path

γ′′(r) = exp(c(r)(JB_{w})) exp(rw)(P_{1})

(note that we flow P_{1} with respect to rB_{w} first). Then γ′′ is a path on M_{0}, γ′′(r) is defined for small values of r, γ′′(0) = P_{1} and the tangent vector to γ′′ is B_{w} + b(γ(r))JB_{w}. So γ′′ coincides with γ′ whenever it is defined.

Next we prove that γ′′(r) is defined for r ≤ t(m). Suppose on γ′ the function τ (see Equation (24)) ranges between the values s_{1} and s_{2}. Then t_{1} < s_{1} and s_{2} < t_{2} (see Proposition-Definition 4.2.1). Pick any r for which γ′′(r) is defined and consider the path exp(tJB_{w}) exp(rw)(P_{1}) for t ranging between 0 and c(r). The function τ is increasing along the path, and on the endpoints its values are between s_{1} and s_{2}. Hence this path lives in the compact set A = τ^{−1}[s_{1}, s_{2}] in M_{0}. From this one can easily deduce that γ′′(r) is well defined for all 0 ≤ r ≤ t(m) and coincides with γ′(r). In particular P_{2} = exp(c(t(m))JB_{w}) exp(t(m)w)(P_{1}). Now P_{2} = ξ(m)P_{1}, hence

τ(P_{2}) = τ(P_{1}) = τ(exp(t(m)w)P_{1})

and τ increases on the trajectories of JB_{w}. So we get that c(t(m)) = 0, i.e., P_{2} = exp(t(m)w)P_{1}. q.e.d.

Now we can prove that ξ is constant on S_{+}. Since ξ is constant on U′ we get from Lemma 4.3.2 that t(m) (see Definition 4.3.2) is a constant t on U′. Let ϕ_{t} be the time t flow of W′ on S_{+}. Consider the map χ = exp(−tw) · ϕ_{t} : S_{+} → S_{+}. S_{+} is a connected real analytic manifold and χ is a real analytic map. Also χ is the identity map on U′. So we deduce that χ is the identity map. So ϕ_{t} is the multiplication by g = exp(tw) on S_{+}. From this we easily deduce that ξ = g on S_{+}.

So ξ assumes a constant value g on S_{+}, hence by (26) it assumes a constant value g^{−1} on S_{−} = −1 · S_{+}. Let Δ = f^{−1}(0). Then Δ is the common boundary of S_{+} and S_{−} in Φ. Since ξ is continuous, we have g = g^{−1}, i.e., g^{2} = 1. q.e.d.

From this we get an immediate corollary

Corollary 4.3.1. The set ξ^{−1}(R′) is everywhere dense in Φ. Here R′ is the set of roots of unity in R.

We are now ready to state and prove our main result:

Theorem 4.3.1. Let M^{2n} (n ≥ 2) be a K-E manifold with positive scalar curvature with an effective structure preserving T^{n}-action. Then precisely one regular orbit L of the T^{n}-action is a minimal Lagrangian submanifold of M. Moreover there is an (n−1)-torus T^{n−1} ⊂ T^{n} and a sequence of non-flat T^{n−1}-invariant immersed minimal Lagrangian tori L_{k} ⊂ M such that L_{k} locally converge to L (in particular the supremum of sectional curvatures of L_{k} and the distance between L and L_{k} goes to 0 as k → ∞).

Proof. Choose a torus T^{n−1} which satisfies the conditions of Lemma 4.1.2. By Corollary 4.3.1 we can choose a sequence of points m_{k} ∈ ξ^{−1}(R′) such that m_{k} converge to a point m in L_{+}(see Definition 4.2.2). The W -trajectories γ_{k} through m_{k} are periodic and live on level sets Σ_{k} of f with Σ_{k} converging to L_{+} in the distance topology. From this we easily see that γ_{k} locally converge to the trajectory L_{+}. One deduces that the immersed minimal Lagrangian tori L_{k} which γ_{k} define as in Lemma 3.3.1 locally converge to the minimal, T -invariant orbit L.

Finally we prove that L_{k} are not flat. From Lemma 4.1.2 we get a vector v in the Lie algebra of T^{n−1} such that the flow vector field X_{v} of v doesn’t have a constant length on Z. The vector field X_{v} along L_{k} is a Killing vector field of L_{k}. To prove that L_{k} is not flat it is enough to prove that |X_{v}|^{2} is non-constant on L_{k}.

The function |X_{v}|^{2} is T^{n}-invariant on Z. Thus it projects to an R-invariant function on M_{red}, i.e., it can be viewed as a function of τ = μ ⊗ w (see Equation (24)) on M_{red}. Also |X_{v}|^{2} is a real analytic function on M_{0} = τ^{−1}(t_{1}, t_{2}). Since |X_{v}|^{2} is non-constant, it is nowhere a locally constant function of τ. We finally claim that the function τ is non-constant on the projection π′(γ_{k}) of γ_{k} onto M_{red}. Indeed if τ were constant on π′(γ_{k}) that means that π′(γ_{k}) is contained in the R-orbit β on M_{red}. But the vector field W (the tangent vector filed to γ_{k}) is horizontal and non-vanishing on S. Hence if π′(γ_{k}) is contained in the R-orbit β on M_{red} that means that π′(γ_{k}) coincides with β. But that by Lemma 3.3.1 means that the preimage ν^{−1}(β) of β in M is a minimal Lagrangian orbit of the T^{n}-action on M and this orbit is different from L since γ_{k} are different from L_{±}. But L = μ^{−1}(0) is the unique minimal Lagrangian T^{n}-orbit in M- a contradiction. Thus |X_{v}|^{2} is non-constant on L_{k} and we are done. q.e.d.

References

[1] M. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14(1) (1982) 1–15.

[2] T. Aubin, Equations du type Monge-Ampere sur les varietes kahleriennes compactes, C. R. Acad. Sci. Paris Sér. A-B 283(3) (1976) A119–A121.

[3] V. Batyrev & E. Selivanova, Einstein-Kähler metrics on symmetric toric Fano manifolds, J. Reine Angew Math. 512 (1999) 225–236.

[4] E. Calabi, Metriques Kähleriennes et fibres holomorphes, Ann. Sci. Ecol. Norn. Sup. (4) 12 (1979) 505–528.

[5] I. Castro & F. Urbano, New examples of minimal Lagrangian tori in the complex projective plane, Manuscripta Math. 85(3) (1994) 265–281.

[6] A. Futaki, Kähler-Einstein metrics and integral invariants, Springer Lecture Notes in Math., 1314, Berlin, 1988.

[7] E. Goldstein, Calibrated Fibrations on Non-Compact manifolds via group actions, Duke Math. Journal 110(2) (2001) 309–344.

[8] E. Goldstein, Calibrated fibrations, to be published in Communications in Analysis and Geometry.

[9] R.Harvey & H.B. Lawson, Calibrated Geometries, Acta Math. 148 (1982) 47–157.

[10] M. Haskins, Special Lagrangian cones, math.DG/0005164.

[11] D. Joyce, Special Lagrangian m-folds in ℂ^{m} with symmetries, math.DG/0008021.

[12] S. Kobayashi, Transformation groups in differential geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, 70, Springer-Verlag, 1972.

[13] Y.-G. Oh, Mean curvature vector and symplectic topology of Lagrangian submanifolds in Einstein-Kähler manifolds, Math. Z. 216 (1994) 471–482.

[14] S. Salamon, Riemannian Geometry and Holonomy Groups, Pitman Research notes in Math., 201, Longmann, Harlow, 1989.

[15] R. Schoen & J. Wolfson, Minimizing volume among Lagrangian submanifolds, Differential Equations: La Pietra 1996 (Florence), Proc. Sympos. Pure Math., 65, Amer. Math. Soc., Providence, 1999, 181–199.

[16] A. Strominger, S.-T. Yau & E. Zaslow, Mirror Symmetry is T-Duality, Nucl. Phys. B476(1-2) (1996) 243–259.

[17] G. Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 137 (1997).

[18] S.-T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation I, Comm. Pure Appl. Math. 31(3) (1978) 339–411.

Stanford University