j. differential geometry 55 (2000) 355-384

QUATERNIONIC MAPS BETWEEN HYPERKÄHLER MANIFOLDS

JINGYI CHEN & JIAYU LI

Abstract

Quaternionic maps (Q-maps) between hyperkähler manifolds are quaternionic analogue of Cauchy-Riemann equations between Kähler manifolds. We provide a necessary and sufficient condition on when a quaternionic map becomes holomorphic with respect to some complex structures in the hyperkähler 2-spheres, and give examples of Q-maps which cannot be holomorphic. When the domain is real 4-dimensional, we analyze the structure of the blow-up set of a sequence of Q-maps, and show that the singular set of a stationary Q-map is H^{1}-rectifiable.

1. Introduction

Quaternionic maps between hyperkähler manifolds arise naturally in higher dimensional gauge theory (cf. [3], [6], [9]). Unlike holomorphic maps, to define quaternionic maps, one needs to use all of the three complex structures, which determine the hyperkähler structures on both the domain and target manifolds. Let M and N be two hyperkähler manifolds and let I, J, K and i, j, k be complex structures on them respectively satisfying the quaternionic identities:

I^{2} = J^{2} = K^{2} = IJK = −Id,
i^{2} = j^{2} = k^{2} = i j k = −Id.

A quaternionic map f : M → N is characterized by

i df I + j df J + k df K = df.

Received October 10, 2000, and, in revised form, January 18, 2001. The first author was supported partially by a fellowship from Alfred P. Sloan Foundation and a NSERC grant, and the second author by the National Science Foundation of China.

In 1935, R. Fueter considered the same equation for f : ℍ → ℍ (cf. [10], [25]) in his effort to generalize complex analysis to the quaternionic setting. Recently, D. Joyce studied ℍ-valued functions defined on hypercomplex manifolds and provided applications to hypercomplex algebraic geometry (cf. [15] and the reference therein). Quaternionic maps automatically minimize the energy functional in their homotopy classes (cf. Proposition 2.2 and [3], [9]), and hence they are harmonic. Since this is a well known property of holomorphic maps between Kähler manifolds, it would be very interesting to know whether quaternionic maps can always be made holomorphic by rotating complex structures or if they constitute a new class of harmonic maps.

In Section 2, we prove Theorem 2.3 which provides a necessary and sufficient condition for a quaternionic map to be holomorphic with respect to some complex structures in the hyperkähler 𝕊^{2} (here we recall that the complex structures compatible with the hyperkähler metric are parameterized by 𝕊^{2}). We also give examples of quaternionic maps which can not be holomorphic with respect to any complex structures.

In Section 3, we analyze the structure of the blow-up set of a sequence of quaternionic maps. It is shown in [17] that the limit map u of a sequence of quaternionic maps u_{k} : M → N with bounded total energy E(u_{k}) ≤ C is a stationary harmonic map, and the blow-up set Σ is stationary. In particular, when M is real 4-dimensional, Σ is 2-rectifiable, so the 2-dimensional Hausdorff measure H^{2}(Σ) is finite and H^{2} almost all points of Σ are contained in a countable union of 2-dimensional C^{1}- submanifolds of M. When M is a compact hyperkähler surface, we show in Proposition 3.3 that the two dimensional components of Σ are a union of minimal surfaces S_{i} away from a closed set whose H^{2}-measure is zero, and moreover if S_{i} ∩ S_{j} contains a curve C, then C ⊂ sing(u). This is achieved by using the quaternionic map equation. The theory describing the interplay between calibrated submanifolds and higher dimensional instantons has been developed in [6] and [26]. We conjecture that the minimal surfaces S_{i} are calibrated by a certain closed 2-form. For an energy minimizing map v, the regularity theory of Schoen and Uhlenbeck [20] asserts that the singular set of v always has Hausdorff dimension ≤ m − 3, where m is the real dimension of the domain. For a stationary harmonic map, the (m − 2)-dimensional Hausdorff measure of its singular set is zero (cf. [2], [7], [12]).

In Section 4, we study the regularity of a stationary quaternionic map u (cf. Definition 4.1) from a hyperkähler surface M. In this section, we assume that the Riemannian metric on N which defines the hyperkähler structure is real analytic. Using a result of Simon [22] on the singular sets of stationary harmonic maps, we prove in Theorem 4.3 that the singular set of u is H^{1}-rectifiable. For energy minimizing maps, this regularity result was obtained in [22]. If the target N does not admit a holomorphic 𝕊^{2} with respect to any complex structure in the hyperkähler 𝕊^{2}, a stationary quaternionic map is smooth outside a finite set of points; if N admits a holomorphic 𝕊^{2} with respect to some complex structure in the hyperkähler 𝕊^{2}, then there exists a stationary quaternionic map whose singular set is a line. Lin’s technique in [18] plays an important role in our analysis about stationary quaternionic maps.

In the last section, we take a different approach. Rather than considering the tangent maps, we show that if there are no holomorphic maps from 𝕊^{2} to N with respect to certain pairs of complex structures on the domain and target spaces, then there is a subsequence, in any sequence of quaternionic maps from a compact hyperkähler surface to N with bounded energy, which converges strongly in W^{1,2} norm to a quaternionic map which is smooth except possibly at finitely many points.

The authors would like to thank Professors W.Y. Ding, R. Schoen, L. Simon, G. Tian and T. Toro for valuable discussions regarding calibrated geometry and stationary harmonic maps and their interests. The authors are grateful to the University of British Columbia, the Academy of Mathematics and System Sciences in Chinese Academy of Sciences, and Massachusetts Institute of Technology, where this work was carried out. Finally, the authors would like to thank the referees for their useful comments.

2. Quaternionic maps

Recall that a hyperkähler manifold is a Riemannian manifold with three covariant constant orthogonal automorphisms I, J and K of the tangent bundle which satisfy the quaternionic identities

I^{2} = J^{2} = K^{2} = IJK = −Id.

For any real numbers a, b, c with a^{2} + b^{2} + c^{2} = 1, we obtain a covariant constant complex structure aI + bJ + cK. We shall refer this 𝕊^{2}-family of complex structures as the hyperkähler 𝕊^{2}. Therefore, SO(3) acts naturally on the covariant constant complex structures. Furthermore, every SO(3) matrix preserves the quaternionic identities. A hyperkähler manifold is of dimension 4k. ℝ^{4k} and the standard 4k dimensional tori naturally carry hyperkähler structures. It is well known that all K3 surfaces are hyperkähler. The moduli space of the irreducible anti-self-dual connections on a K3 surface is hyperkähler as well (cf. [14], [19] and other people’s work). There is also construction via moment maps by Hitchin and others (cf. [13] and the reference therein).

Definition 2.1. Let M and N be two hyperkähler manifolds with complex structures J^{α} and J^{β} respectively for α, β = 1, 2, 3 which satisfy the quaternionic identities. A smooth map u : M → N is called a quaternionic map if

A_{αβ}J^{β} ∘ du ∘ J^{α} = du, (1)

where A_{αβ} denote the entries of a matrix A in SO(3).

It can be verified (cf. [3]) that holomorphic and anti-holomorphic maps with respect to some complex structures in the hyperkähler 𝕊^{2} on M and N are quaternionic maps. In particular, the identity map from M to itself satisfies

I du I − J du J − K du K = du.

Note that the coefficients in this equation form a SO(3)-matrix, i.e., the diagonal matrix with diagonal elements 1, −1, −1, hence the identity map is quaternionic.

In local coordinates, the equation (1) reads as

A_{αβ}(J^{α})_{i}^{j}(J^{β})_{m}^{n}∂_{j}u^{m} = ∂_{i}u^{n}.

Here and in sequel, we sum up all repeated upper and lower indices. This equation is a quaternionic analog of the Cauchy-Riemann equation defining holomorphic maps. Since SO(3) preserves the quaternionic identities, we can always choose complex structures J^{α} for M and J^{β} for N such that A_{αβ} = δ_{αβ} in (1). In the sequel, we shall assume that A_{αβ} = δ_{αβ}.

Let g and h be the Riemannian metrics on M and N respectively. Consider the energy functional

E(u) = \frac{1}{2}∥du∥^{2} = \frac{1}{2} ∫_{M} g^{ij}h_{mn}∂_{i}u^{m}∂_{j}u^{n}dV,

where dV is the volume element, and the functional

E_{T} (u) = ∫_{M} ∑_{α}⟨J^{α}, u^{∗}J^{α}⟩dV
= \frac{1}{2} ∫_{M} ∑_{α}(J^{α})^{pq}J_{mk}^{α}∂_{p}u^{m}∂_{q}u^{k}dV,

and set

I(u) = \frac{1}{2} ∫_{M} |du − ∑_{α} J^{α} ∘ du ∘ J^{α}|^{2}dV.

It is clear that I(u) = 0 if and only if u is a quaternionic map. Let J be a complex structure on M and let J be a complex structure on N. That holomorphic maps between Kähler manifolds are energy minimizers in their homotopy classes follows easily from

E(u) + ∫_{M}⟨J, u^{∗}J ⟩dV = \frac{1}{4} ∫_{M} |du − J ∘ du ∘ J|^{2}dV. (2)

Note that ∫_{M}⟨J, u^{∗}J ⟩dV is just ∫_{M}⟨ω_{J}, u^{∗}ω_{J} ⟩dV where ω_{J} is the Kähler form on M defined by ω_{J}(X, Y) = g(JX, Y) and ω_{J} is the Kähler form on N with respect to J. In order to investigate similar properties for quaternionic maps, we have (cf. [3], [17], [9]):

Proposition 2.2. For any smooth map u : M → N, we have

E(u) + E_{T} (u) = \frac{1}{4}I(u). (3)

If u is a quaternionic map, then it minimizes energy in its homotopy class.

Proof. It is clear that

I(u) = \frac{1}{2} ∫_{M} g^{ij}h_{mn}(∂_{i}u^{m} − ∑_{α}(J^{α})_{i}^{p}(J^{α})_{k}^{m}∂_{p}u^{k})

(∂_{j}u^{n} − ∑_{α}(J^{α})_{j}^{q}(J^{α})_{k}^{n}∂_{q}u^{k})dV

= E(u) − ∫_{M} ∑_{α}(J^{α})^{pq}J_{km}^{α}∂_{p}u^{m}∂_{q}u^{k}dV

+\frac{1}{2} ∫_{M} ∑_{α,γ} g^{ij}h_{mn}(J^{α})_{i}^{p}(J^{α})_{k}^{m}(J^{γ})_{j}^{q}(J^{γ})_{l}^{n}∂_{p}u^{k}∂_{q}u^{l}dV.

We claim that

∑_{α,γ} g^{ij}h_{mn}(J^{α})_{i}^{p}(J^{α})_{k}^{m}(J^{γ})_{j}^{q}(J^{γ})_{l}^{n} = 3δ_{pq}δ_{kl} + 2 ∑_{α}(J^{α})^{pq}(J^{α})_{kl}.

It suffices to prove the above identity in the normal coordinates. Take a system of normal coordinates around any point p ∈ M and a system of normal coordinates around u(p) ∈ N. Then we have

∑_{α,γ} g^{ij}h_{mn}(J^{α})_{i}^{p}(J^{α})_{k}^{m}(J^{γ})_{j}^{q}(J^{γ})_{l}^{n}

= ∑_{α,γ} J_{ip}^{α}J_{km}^{α}J_{iq}^{γ}J_{lm}^{γ}
= ∑_{α,γ} J_{pi}^{α}J_{iq}^{γ}J_{km}^{α}J_{ml}^{γ}
= 3δ_{pq}δ_{kl} + 2 (J_{pq}^{1}J_{kl}^{1} + J_{pq}^{2}J_{kl}^{2} + J_{pq}^{3}J_{kl}^{3}) ,

and the claim follows. Then it is clear that

I(u) = E(u) + 2E_{T} (u) + 3E(u) + 2E_{T} (u)
= 4(E(u) + E_{T} (u)).

This proves (3). For fixed complex structures J^{α} and J^{β}, E_{T} only depends on the homotopy class of u (cf. [16]). Therefore, if u is a quaternionic map, the right side of (3) vanishes and in turn u is an energy minimizer in its homotopy class. q.e.d.

We now establish a criterion which detects when a quaternionic map becomes holomorphic with respect to some complex structures in the hyperkähler 𝕊^{2}.

Theorem 2.3. Suppose that u is a quaternionic map. Let A be a 3 × 3-matrix whose (α, β)-entries are − ∫_{M}⟨J^{α}, u^{∗}J^{β}⟩dV for α, β = 1, 2, 3. Then

(trA)^{2} ≥ max{eigenvalues of AA^{t}}

and the equality holds if and only if u is a holomorphic map with respect to some complex structures in the hyperkähler 𝕊^{2} on M and on N.

Proof. Setting J = X_{α}J^{α} with |X| = 1 and J = Y_{β}J^{β} with |Y | = 1, then from (2) we have

E(u) = XAY^{t} + \frac{1}{4} ∫_{M} |du − J ∘ du ∘ J|^{2}dV.

Since u is a quaternionic map, by (3) we have

E(u) = trA.

It follows that

trA ≥ XAY^{t} (4)

for any unit vectors X and Y, and in fact the equality holds if and only if u is holomorphic with respect to J and J. The eigenvalues of AA^{t} are all nonnegative. If 4λ^{2} is an eigenvalue of the real symmetric 3 × 3 matrix AA^{t} where λ ≥ 0, there is a unit vector Y_{λ} in ℝ^{3} such that

AA^{t}Y_{λ}^{t} = 4λ^{2}Y_{λ}^{t} ,

and hence

Y_{λ}(A^{t}AY_{λ}^{t}) = Y_{λ}(AA^{t}Y_{λ}^{t}) = 4λ^{2}Y_{λ}Y_{λ}^{t}.

In turn, we have |AY_{λ}^{t}| = 2λ. If λ ≠ 0, we choose X_{λ}^{t} = \frac{1}{2λ}AY_{λ}^{t} and get

X_{λ}AY_{λ}^{t} = 2λ.

We therefore have trA ≥ 2λ and consequently

(trA)^{2} ≥ max{eigenvalues of AA^{t}}.

This is trivially true if all of the eigenvalues of AA^{t} are 0 since A is the zero matrix in this case. This proves the first part of the theorem.

As for the second part, we introduce the Lagrange multiplier:

F (X, Y) = XAY^{t} − λ(|X|^{2} − 1) − μ(|Y |^{2} − 1).

If XAY^{t} attains its maximum at two unit vectors V, W ∈ ℝ^{3}, we have F_{X} = F_{Y} = 0 at X = V, Y = W. This leads to

AW^{t} = 2λV^{t} and V A = 2μW.

It follows that

2λ = 2λ|V |^{2} = V AW^{t} = 2μ|W |^{2} = 2μ.

This implies

A^{t}AW^{t} = 2λA^{t}V^{t} = 4λμW^{t} = 4λ^{2}W^{t}.

Therefore 4λ^{2} is an eigenvalue of A^{t}A.

If u is holomorphic with respect to some complex structures X_{α}J^{α} on M and Y_{α}J^{α} on N in the hyperkähler 𝕊^{2}, then XAY^{t} attains the maximum trA by (4). The discussion above asserts that trA = XAY^{t} = 2λ and 4λ^{2} is an eigenvalue of A^{t}A, so (trA)^{2} = max{eigenvalues of AA^{t}} according to the first part of the theorem.

Conversely, if (trA)^{2} = max{eigenvalues of AA^{t}} > 0, we set 2λ = trA, then 4λ^{2} is an eigenvalue of A^{t}A. Suppose that |Y | = 1 and

A^{t}AY^{t} = 4λ^{2}Y^{t}.

Then we have Y A^{t}AY^{t} = 4λ^{2}, and hence |AY^{t}|^{2} = 4λ^{2}. We choose X^{t} = \frac{1}{2λ}AY^{t} and we get XAY^{t} = 2λ = trA. So by (4) u is a holomorphic map with respect to J = X_{α}J^{α} and J = Y_{α}J^{α}. If A is the zero matrix, the quaternionic map u is constant. q.e.d.

Corollary 2.4. Let Ω be an open domain in M with smooth boundary ∂Ω. Suppose that u is a quaternionic map from Ω to N which extends smoothly to \overline{Ω} → N. Let A be a 3 × 3-matrix whose (α, β)-entries are − ∫_{Ω}⟨J^{α}, u^{∗}J^{β}⟩dV for α, β = 1, 2, 3. Then

(trA)^{2} ≥ max{eigenvalues of AA^{t}}

and the equality holds if and only if u is a holomorphic map with respect to some complex structures in the hyperkähler 𝕊^{2} on M and on N.

Proof. If Ω is a domain in M with smooth boundary and u : Ω → N is a smooth map, we still have

E(u) + E_{T} (u) = \frac{1}{4}I(u).

Now E_{T} (u) is a homotopy invariant among maps v : Ω → N which are homotopic to u relative to ∂Ω with v|_{∂Ω} = u|_{∂Ω} (i.e., there exists a continuous family of maps u_{t}, 0 ≤ t ≤ 1, with u_{0} = u, u_{1} = v and u_{t} ≡ u on ∂Ω). To see this, we observe that the pull-back 2-forms of the Kähler form ω_{J^{β}} by u and v stay in the same cohomology class H^{2}(Ω, rel ∂Ω):

v^{∗}ω_{J^{β}} − v^{∗}ω_{J^{β}} = dη

for some 1-form η on Ω and η(X) = 0 for any vector X tangent to ∂Ω.

It follows that

∫_{Ω}⟨ω_{J^{α}}, v^{∗}ω_{J^{β}}⟩ dV − ∫_{Ω}⟨ω_{J^{α}}, u^{∗}ω_{J^{β}}⟩ dV = ∫_{Ω}⟨ω_{J^{α}}, dη⟩ dV
= ∫_{Ω} dη ∧ \frac{ω_{J^{α}}^{m−1}}{(m − 1)!}
= ∫_{Ω} d (η ∧ \frac{ω_{J^{α}}^{m−1}}{(m − 1)!})

= ∫_{∂Ω} η ∧ \frac{ω_{J^{α}}^{m−1}}{(m − 1)!}
= 0

by Stokes’ theorem and η vanishes along ∂Ω. The rest of the argument is the same as that in the proof of Theorem 2.3. q.e.d.

Assume the real dimensions of M and N are four. To write (1) in local coordinates, we choose a coordinate system at a point x in M and a coordinate system at f(x) in N, so that the matrix expressions of the complex structures take the following form (cf. [14]): let id be the 2 × 2 identity matrix and let I be the matrix for the standard complex structure on ℂ, then we take

\begin{matrix}J^{1} = J^{1} = ( \begin{matrix}0 & I \\ I & 0 \end{matrix} ) , J^{2} = J^{2} = ( \begin{matrix}I & 0 \\ 0 & −I \end{matrix} ) , \\ J^{3} = J^{3} = ( \begin{matrix}0 & id \\ −id & 0 \end{matrix} ) . \end{matrix} (5)

Denote the differential du of u by the matrix (\frac{∂u^{α}}{∂x^{i}} ) for α, i = 1, 2, 3, 4. Set

u_{i} = (\frac{∂u^{1}}{∂x^{i}} , \frac{∂u^{2}}{∂x^{i}} , \frac{∂u^{3}}{∂x^{i}} , \frac{∂u^{4}}{∂x^{i}} )^{t} .

Simple computation then shows that the quaternionic equation (1) in dimension four is equivalent to

u_{1} − J^{2}u_{2} + J^{3}u_{3} − J^{1}u_{4} = 0, (6)

which leads to

\{ \begin{matrix}u_{1}^{1} + u_{2}^{2} + u_{3}^{3} + u_{4}^{4} & = & 0 \\ u_{1}^{2} − u_{2}^{1} + u_{3}^{4} − u_{4}^{3} & = & 0 \\ u_{1}^{3} − u_{3}^{1} − u_{2}^{4} + u_{4}^{2} & = & 0 \\ u_{1}^{4} − u_{4}^{1} − u_{3}^{2} + u_{2}^{3} & = & 0. \end{matrix} (7)

If dim_{ℝ} M = 4 and dim_{ℝ} N = 4n, to write (1) in local coordinates, we choose a coordinate system {x_{1}, x_{2}, x_{3}, x_{4}} around x and a coordinate system around u(x) in N, so that the matrix expressions of the complex structures take the following form: for each α = 1, 2, 3, J^{α} is as before and J^{α} has n copies of J^{α} along the diagonal and 0 elsewhere in block form. Then the quaternionic equation is given by

\{ \begin{matrix}u_{1}^{4i−3} + u_{2}^{4i−2} + u_{3}^{4i−1} + u_{4}^{4i} & = & 0 \\ u_{1}^{4i−2} − u_{2}^{4i−3} + u_{3}^{4i} − u_{4}^{4i−1} & = & 0 \\ u_{1}^{4i−1} − u_{3}^{4i−3} − u_{2}^{4i} + u_{4}^{4i−2} & = & 0 \\ u_{1}^{4i} − u_{4}^{4i−3} − u_{3}^{4i−2} + u_{2}^{4i−1} & = & 0 \end{matrix} (8)

for 1 ≤ i ≤ n.

Now we provide examples of quaternionic maps.

Example 2.5. Let M and N be ℝ^{4} with complex structures J^{α} = J^{α} which are defined by (5). Let u : ℝ^{4} → ℝ^{4} defined by

u(x_{1}, x_{2}, x_{3}, x_{4}) = (0, x_{1}x_{2}, −x_{1}x_{3}, −x_{2}x_{3}).

Using (7), one can check that u is a quaternionic map. Direct computation leads to

Du = ( \begin{matrix}0 & 0 & 0 & 0 \\ x_{2} & x_{1} & 0 & 0 \\ −x_{3} & 0 & −x_{1} & 0 \\ 0 & −x_{3} & −x_{2} & 0 \end{matrix} ) .

Since

det ( \begin{matrix}x_{2} & x_{1} & 0 \\ −x_{3} & 0 & −x_{1} \\ 0 & −x_{3} & −x_{2} \end{matrix} ) = −x_{1}x_{2}x_{3},

we conclude that the rank of Du is three at (x_{1}, x_{2}, x_{3}, x_{4}) as long as x_{1}x_{2}x_{3} ≠ 0. However, the rank of a holomorphic map should be even. It follows that u can not be a holomorphic map with respect to any complex structures on ℝ^{4}.

Example 2.6. Consider a map u : (0, 1)×(0, 1)×(0, 1)×(0, 1) → ℝ^{4} given by

u(x_{1}, x_{2}, x_{3}, x_{4}) = (a_{1}x_{1}, a_{2}x_{2}, a_{3}x_{3}, a_{4}x_{4}),

where a_{i} ∈ ℝ for i = 1, 2, 3, 4. By (7), if

a_{1} + a_{2} + a_{3} + a_{4} = 0,

then u is a quaternionic map. Suppose that the elements of the matrix A is A_{αβ} (α, β = 1, 2, 3). We have

A_{αβ} = −\frac{1}{2} ∑_{p,q=1}^{4} a_{p}a_{q}(J^{α})^{pq}(J^{β})_{pq}.

A simple computation shows that

A = − ( \begin{matrix}a_{1}a_{4} + a_{2}a_{3} & 0 & 0 \\ 0 & a_{1}a_{2} + a_{3}a_{4} & 0 \\ 0 & 0 & a_{1}a_{3} + a_{2}a_{4} \end{matrix} ) .

Choosing a_{2} = a_{3} = a_{4} = μ > 0 and a_{1} = −3μ. Then u is a quaternionic map. But

A = 2 ( \begin{matrix}μ^{2} & 0 & 0 \\ 0 & μ^{2} & 0 \\ 0 & 0 & μ^{2} \end{matrix} ) .

So trA = 6μ^{2} and

AA^{t} = 4 ( \begin{matrix}μ^{4} & 0 & 0 \\ 0 & μ^{4} & 0 \\ 0 & 0 & μ^{4} \end{matrix} ) .

Applying Corollary 2.4, we see that u is not a holomorphic map with respect to any complex structures on (0, 1) × (0, 1) × (0, 1) × (0, 1) and ℝ^{4}.

Example 2.7. Let 𝕋^{4} = 𝕊^{1} × 𝕊^{1} × 𝕊^{1} × 𝕊^{1} be the standard 4-dimensional torus. Consider a map u : 𝕋^{4} → 𝕋^{4} determined by

u(e^{2πix_{1}}, e^{2πix_{2}}, e^{2πix_{3}}, e^{2πix_{4}}) = (e^{2πix_{1}}, e^{2πix_{2}}, e^{2πix_{3}}, e^{−6πix_{4}}).

In local coordinates,

u(x_{1}, x_{2}, x_{3}, x_{4}) = (x_{1}, x_{2}, x_{3}, −3x_{4}).

It is clear from (7) that u is a quaternionic map. By an argument similar to the one used in the second example, we conclude that u is not holomorphic with respect to any complex structures on 𝕋^{4} by Theorem 2.3.

3. The blow-up set

Let M and N be two compact hyperkähler manifolds. Let m = dim_{ℝ} M. Suppose that u_{k} is a sequence of smooth quaternionic maps with E(u_{k}) ≤ C. We recall that as a sequence of harmonic maps with bounded energy the blow-up set of u_{k} can be defined as

Σ = ⋂_{r>0}{x ∈ M| lim inf_{k→∞} r^{2−m} ∫_{B_{r}(x)} | ▽ u_{k}|^{2}dy ≥ ϵ_{0}}.

We can always assume that u_{k} ⇀ u weakly in W^{1,2}(M, N) and that

| ▽ u_{k}|^{2}dx ⇀ | ▽ u|^{2}dx + ν,

in the sense of measure as k → ∞. Here ν is a nonnegative Radon measure on M with support in Σ. It is known that Σ is a closed H^{m−2}-rectifiable set, so we can define the gradient operator ▽^{Σ} (cf. Section 12.1 in [23]) and the divergence along Σ

div_{Σ}X = ∑_{j=1}^{m−2}(▽_{e_{j}}^{Σ}X) · e_{j}

for any smooth vector field X on M, where e_{1}, ⋯, e_{m−2} is any orthonormal basis for T_{x}Σ, x ∈ Σ (cf. Section 16 in [23]). We say that Σ is stationary if for any smooth vector field X on M, we have

∫_{Σ} div_{Σ}Xν = 0.

A map f from M to N is said to be a stationary harmonic map, if it is a weakly harmonic map and for any smooth vector field X on M, we have

∫_{M} (| ▽ f|^{2}div(X) − 2⟨df(▽_{α}X), df(\frac{∂}{∂x^{α}})⟩) dV = 0.

The following result is proved in [17] (Theorem 4.4 and Theorem 4.5 in [17]).

Theorem 3.1 ([17]). Let M and N be two compact hyperkähler manifolds. Suppose that u_{k} is a sequence of quaternionic maps with bounded energy E(u_{k}) ≤ C. Then by taking a subsequence if necessary, u_{k} ⇀ u weakly in W^{1,2}(M, N) and the limit map u is a stationary harmonic map which is also a stationary quaternionic map. Moreover, the blow-up set Σ is stationary and has no boundary.

From now on, we assume that M is a hyperkähler surface, that is m = 4.

Since Σ is stationary, it is a union of smooth real 2-dimensional surfaces outside a closed set Σ_{0} with H^{2}(Σ_{0}) = 0.

Definition 3.2. Let B_{r}(x) be a geodesic ball and let

Σ_{1} = { x ∈ Σ | H^{2}(Σ ∩ B_{r}(x)) > 0 for any r > 0}.

We call Σ_{1} the two dimensional component of Σ.

It is clear that if x ∈ Σ \ Σ_{1}, there is r_{0} > 0 such that H^{2}(Σ∩B_{r_{0}}(x)) = 0. Consequently, u_{k} → u strongly in W^{1,2}(B_{r_{0}}(x), N).

Proposition 3.3. Let M be a compact hyperkähler surface and let N be a compact hyperkähler manifold. Suppose that u_{k} is a sequence of smooth quaternionic maps with bounded energy E(u_{k}) ≤ C. Let Σ_{1} be the two dimensional component of the blow-up set Σ for the sequence. Then Σ_{1} \Σ_{0} = ∪_{i}S_{i} where S_{i} are minimal surfaces. Let u be the weakly limiting map of the sequence u_{k}. If S_{i} and S_{j} meets transversely along a curve C, then C ⊂ sing(u).

Proof. By the constancy theorem for stationary varifolds (cf. [1], [23]), the density function of Σ is constant on each connected component. Then the first part of the proposition is just a re-statement of Theorem 3.1. It suffices to show that, if S_{i} and S_{j} meets transversely along a curve C, then C ⊂ sing(u). Since each S_{i} is smooth, we can choose three tangent vectors e′_{1}, e′_{2}, e′_{3} ∈ T_{x}S_{i} ∪ T_{x}S_{j} for any x ∈ C. If C ⊄ sing(u), we may choose x to be the point where u is smooth. So u is smooth in a neighborhood of x. By Lemma 2.2 in [17], the nonnegative Radon measure ν_{e′_{i}} = 0 and in the measure convergence |∇_{e′_{i}}u_{k}|^{2}dV ⇀ |∇_{e′_{i}}u|^{2}dV + ν_{e′_{i}} for all tangential direction e′_{i}, i = 1, 2, 3, therefore we have

lim_{r→0} lim_{k→∞} r^{−2} ∫_{B_{r}(x)} | ▽_{e′_{i}} u_{k}|^{2}dV = lim_{r→0} r^{−2} ∫_{B_{r}(x)} | ▽_{e′_{i}} u|^{2}dV
= 0,

for i = 1, 2, 3, here the last equality follows from the smoothness of u at x. We choose an orthonormal frame {e_{1}, e_{2}, e_{3}, e_{4}} on the tangent bundle T_{U}M over a neighborhood U of x in M such that e_{i} = ∑_{j=1}^{3} A_{ij}e′_{j} at x, for i = 1, 2, 3, where A = (A_{ij}) is an invertible 3 × 3-matrix. So we have

lim_{r→0} lim_{k→∞} r^{−2} ∫_{B_{r}(x)} ∑_{i=1}^{3} | ▽_{e_{i}} u_{k}|^{2}dV = 0.

By the quaternionic map equation, we get

lim_{r→0} lim_{k→∞} r^{−2} ∫_{B_{r}(x)} | ▽_{e_{4}} u_{k}|^{2}dV = 0.

This implies

lim_{r→0} lim_{k→∞} r^{−2} ∫_{B_{r}(x)} | ▽ u_{k}|^{2}dV = 0,

which contradicts that x ∈ Σ is a blow-up point. q.e.d.

4. Removable singularity

In this section, we prove that the singular set of a stationary quaternionic map defined in [17] is a one dimensional rectifiable set.

Definition 4.1. Let M and N be two hyperkähler manifolds. A map u from M to N is a stationary quaternionic map if (1) u ∈ W^{1,2}(M, N) and (2) u is a smooth quaternionic map outside a singular set Σ which is of Hausdorff codimension at least two.

It is proved in [17] that a stationary quaternionic map is a stationary harmonic map. Let us recall some basic terminologies and facts about stationary harmonic maps. Let u(x) be a stationary harmonic map from M to N. The regular set reg (u) of u is defined as the set of points x ∈ M such that u is smooth in some neighborhood of x. It is clear that reg(u) is open in M. The singular set sing(u) of u is the complement of reg(u). The density function Θ_{u} of u is defined by

Θ_{u}(x) = lim_{r→0} r^{2−m} ∫_{B_{r}(x)} | ▽ u|^{2}dV,

where m is the real dimension of M. The monotonicity inequality for u guarantees that the limit exists, and the density function Θ_{u} of u is upper semi-continuous. It is proved (cf. Theorem I.4 in [2], and [7]) that x ∈ reg(u) if and only if Θ_{u}(x) = 0. To study the local behavior of the map u around a point x ∈ M, we take a small convex geodesic ball B(x) centered at x in M. For any y ∈ B(x), there is a unique geodesic γ(x, y) which connects x and y with unit speed. We shall denote the point on γ(x, y) with distance r from x by x + ry for any r ∈ [0, 1). At each singular point, the tangent maps exist:

Theorem 4.2 ([17]). Let u be a stationary harmonic map from M to N. Assume that x ∈ sing(u) and set u_{x,r}(y) = u(x + ry). Then for any sequence r_{i′} → 0 there is a subsequence r_{i} → 0 such that u_{x,r_{i}} ⇀ ϕ weakly in W^{1,2}(ℝ^{m}, N) and

| ▽ u_{x,r_{i}}|^{2}dy ⇀ | ▽ ϕ|^{2}dy + θ(y)H^{m−2}⌊ Σ_{x}

in the sense of measure. Moreover,

(1) ϕ(λy) = ϕ(y) for all λ > 0 and y ∈ ℝ^{m}, i.e., ϕ(y) is of homogeneous degree zero.

(2) θ(λy) = θ(y) for all λ > 0 and y ∈ Σ_{x}, i.e., θ(y) is of homogeneous degree zero.

(3) Σ_{x} is a tangent cone, which means λΣ_{x} = Σ_{x} for all λ > 0.

The map ϕ obtained in this theorem is called a tangent map of u at x. Let x ∈ M and take a sequence of points x_{i} → x with Θ_{u}(x_{i}) ≥ Θ_{u}(x). We define

u_{x_{i},r}(y) = u(x_{i} + ry).

By a similar argument as in the proof of Theorem 4.2, we can show that there is a subsequence r_{i} → 0 such that u_{x_{i},r_{i}} ⇀ ϕ weakly in W^{1,2}(ℝ^{m}, N) and

| ▽ u_{x_{i},r_{i}}|^{2}dy ⇀ | ▽ ϕ|^{2}dy + θ(y)H^{m−2}⌊ Σ_{x}

in the sense of measure. ϕ(y) and θ(y) are of homogeneous degree zero and Σ_{x} is a tangent cone. We call such a map ϕ the pseudo-tangent map of u at x as in [22]. If u is a quaternionic map, by Theorem 3.1 and Theorem 4.2 we know that ϕ(y) is a stationary harmonic map and θ(y) is constant in each connected domain in Σ_{x} by the constancy theorem (cf. [23] Theorem 41.1).

Theorem 4.3. Let u be a stationary quaternionic map from a hyperkähler surface M to a hyperkähler manifold N with a real analytic metric which defines the hyperkähler structure. Then the Hausdorff dimension of the singular set of u is at most one, and if it is one, the singular set is rectifiable.

Proof. Since u is a stationary harmonic map by Theorem 4.4 in [17], it is smooth outside a closed subset in M with H^{2} measure zero.

There is a very useful result in Section 9 of [22] about the singular set of stationary harmonic maps. Simon shows the following: Let u be a stationary harmonic map from a m-dimensional domain Ω with a smooth metric to a Riemannian manifold with a real analytic metric. If all tangent map and all pseudo-tangent maps ϕ of u are stationary for the energy functional, and if the set of all such ϕ with density Θ_{ϕ}(0) ≤ β (for any given β) lies in a compact set (relative to the local L^{2} metric) of stationary maps all with singular set of dimension ≤ m_{1}, then sing(u) is locally a finite union of locally m_{1}-rectifiable locally compact subsets. In general, unlike the minimizing maps, neither the stationary assumption on tangent maps and pseudo-tangent maps nor the compactness is known to hold automatically if u is stationary.

We only prove the case that N is also a hyperkähler surface, the proof for the general case is the same.

Step 1. We first show the compactness of the pseudo-tangent maps of u.

Suppose that u_{k} is a sequence of pseudo-tangent maps of u. We can always assume that u_{k} ⇀ u weakly in W_{loc}^{1,2}(ℝ^{4}, N) and that |▽u_{k}|^{2}dx ⇀ |▽u|^{2}dx+ν in the sense of measure as k → ∞. Here ν is a nonnegative Radon measure on ℝ^{4} with support in Σ, and Σ is the blow-up set of the sequence u_{k}. It is clear that Σ is a stationary tangent cone. In particular, we know that the origin 0 of ℝ^{4} belongs to Σ and θ is constant. We will prove H^{2}(Σ) = 0 by deriving a contradiction if H^{2}(Σ) > 0.

Because Σ is H^{2}-rectifiable, it decomposes into

Σ = ⋃_{i=1}^{∞} Σ_{i}, Σ_{i} ∩ Σ_{j} = ∅

if i ≠ j, where H^{2}(Σ_{0}) = 0, Σ_{i} is H^{2}-rectifiable and Σ_{i} ⊂ M_{i} with M_{i} an embedded 2-dimensional C^{1} submanifold of ℝ^{4} for all i ≥ 1; and T_{x}Σ = T_{x}M_{i} for H^{2}-a.e. x ∈ Σ_{i} (cf. p.61 in [23]). Suppose that (x^{1}, x^{2}, x^{3}, x^{4}) is a coordinate system around 0 such that

\frac{∂}{∂x^{1}} , \frac{∂}{∂x^{2}} ∈ T (Σ)

for H^{2}-a.e. x ∈ Σ. We recall the following result in [17]: if T = ∑_{α=1}^{2} ξ^{α} \frac{∂}{∂x^{α}} then

lim_{ϵ→0} lim_{k→∞} ∫_{B_{ϵ}(Σ)} | ▽_{T} u_{k}|^{2}dV = 0. (9)

Here and in the sequel we denote by B_{ϵ}(Σ) = {x ∈ M| dist (x, Σ) < ϵ}. Set

F_{kϵ}(x) = ∫_{B_{ϵ}^{2}(0)} | ▽_{T} u_{k}|^{2}(x, x′)dx′

for x ∈ Σ. Here and in the sequel we denote by B_{r}^{2}(x) the metric ball centered at x with radius r in ℝ^{2}. We consider the Hardy-Littlewood maximal function M_{F_{kϵ}}(x) of F_{kϵ}(x) (cf. [24]), which is defined by

M_{F_{kϵ}}(x) = sup_{0<r<1} r^{−2} ∫_{B_{r}^{2}(x)} F_{kϵ}(x)dx.

By the weak type (1, 1) inequality (cf. [24]) for M_{F_{kϵ}}(x), we have

H^{2}\{x ∈ Σ|M_{F_{kϵ}}(x) ≥ λ\} ≤ \frac{C}{λ} ∫_{B_{ϵ}(Σ)} ∑_{α=1}^{2} | ▽_{α} u_{k}|^{2}dV

for any λ > 0. By (9), we have

lim_{ϵ→0} lim_{k→∞} H^{2}{x ∈ Σ|M_{F_{kϵ}}(x) ≥ λ} = 0.

Thus, for any integer l > 0,

H^{2}( ⋃_{n_{0}=1}^{∞} ⋂_{n=n_{0}}^{∞} ⋃_{k_{0}=1}^{∞} ⋂_{k=k_{0}}^{∞} \{x ∈ Σ | M_{F_{k( \frac{1}{n})}}(x) ≥ \frac{1}{l} \} ) = 0.

By the partial regularity result for stationary harmonic maps in [2] and [7], we can find x_{l} ∈ Σ ⊂ B_{1}^{2}(0) × {0}, such that for any n_{0} > 0 and any k_{0} > 0 there are n_{l} > n_{0} and k_{l} > k_{0} such that

x_{l} → 0 and M_{F_{k_{l}( \frac{1}{n_{l}} )}}(x_{l}) < \frac{1}{l} (10)

and u_{k} is smooth near (x_{l}, x′) for all x′ ∈ B_{1}^{2}(0). We claim that for all k sufficiently large there exist δ_{k} → 0 such that

max_{x′∈B_{1}^{2}(0)} δ_{k}^{−2} ∫_{B_{δ_{k}}^{2} (x_{k})×B_{δ_{k}}^{2} (x′)} | ▽ u_{k}|^{2}(x, x′)dxdx′ = \frac{ϵ_{0}}{8 · 2^{8}} , (11)

and

\frac{|x_{k}|}{δ_{k}} ≤ C, (12)

where ϵ_{0} is the small constant in the small energy regularity theorem (cf. [2], [7]). In fact, since u_{k}(x) is smooth at (x_{k}, x′), for any given k and for δ < δ(k), we have

δ^{−2} ∫_{B_{δ}^{2}(x_{k})×B_{δ}^{2}(x′)} | ▽ u_{k}|^{2}(x, x′)dxdx′ ≤ \frac{ϵ_{0}}{16 · 2^{8}} .

On the other hand, since x ∈ Σ, for fixed δ > 0 and sufficiently large k,

δ^{−2} ∫_{B_{δ}^{2}(x_{k})×B_{δ}^{2}(0)} | ▽ u_{k}|^{2}(x, x′)dxdx′ ≥ \frac{ϵ_{0}}{4 · 2^{8}} .

Therefore we can choose δ_{k} > 0 so that (11) and (12) hold.

By (10), (11) and (12), we can find ϵ_{k} → 0, r_{k} → 0, (x_{k}, x′_{k}) ∈ Σ with

lim_{k→∞} \frac{(x_{k}, x′_{k})}{r_{k}} = p = (p^{1}, p^{2}, p^{3}, p^{4}), (13)

where |p^{1}| < ∞, |p^{2}| < ∞, |p^{3}| ≤ ∞, |p^{4}| ≤ ∞, and a subsequence of u_{k}, which we also denote by u_{k} for simplicity, such that

r_{k}^{−2} ∫_{B_{r_{k}}^{2} (x_{k})×B_{r_{k}}^{2} (x′_{k})} | ▽ u_{k}|^{2}(x, x′)dxdx′ = \frac{ϵ_{0}}{8 · 2^{8}} ,

M_{F_{kϵ_{k}}} (x_{k}, x′_{k}) < \frac{1}{k}

and

lim_{k→∞} \frac{ϵ_{k}}{r_{k}} = ∞.

We define

v_{k}(y) = u_{k}((x_{k}, x′_{k}) + r_{k}y).

It is obvious that

∫_{B_{1}^{2}(0)×B_{1}^{2}(0)} | ▽ v_{k}|^{2}(x, x′)dxdx′ = \frac{ϵ_{0}}{8 · 2^{8}} (14)

and

sup_{0<r<1}(\frac{r}{r_{k}})^{−2} ∫_{B2 ; \frac{r}{r_{k}} (0)×B2 ; \frac{ϵ_{k}}{r_{k}} (0)} ∑_{i=1}^{2} | ▽_{i} v_{k}|^{2}dV < \frac{1}{k} . (15)

By the monotonicity inequality for stationary harmonic maps, for any R > 0, we have

R^{−2} ∫_{B_{R}^{2}(0)×B_{R}^{2}(0)} | ▽ v_{k}|^{2}dV ≤ C.

By the diagonal subsequence argument, we obtain a subsequence of v_{k}, still denoted by v_{k}, such that v_{k} ⇀ v weakly in W^{1,2}(ℝ^{4}, N), and v is a weak harmonic map. It follows that

R^{−2} ∫_{B_{R}^{2}(0)×B_{R}^{2}(0)} | ▽ v|^{2}dV ≤ C. (16)

Since r_{k} → 0 and ϵ_{k}/r_{k} → ∞ as k → ∞, we get by (15) that

∑_{i=1}^{2} ∫_{B_{R}^{2}(0)×B_{R}^{2}(0)} | ▽_{i} v|^{2}dV = 0

for any R > 0, hence ▽_{i}v ≡ 0 for i = 1, 2. By (16), we have

∑_{i=3}^{4} ∫_{B_{R}^{2}(0)} | ▽_{i} v|^{2}dV ≤ C,

where C does not depend on R. Therefore, we have

∑_{i=3}^{4} ∫_{ℝ^{2}} | ▽_{i} v|^{2}dV ≤ C.

By the removable singularity theorem in [21], we can extend v to a smooth harmonic map from 𝕊^{2} to N.

Let ϕ(y) ∈ C_{0}^{∞}(B_{1}^{2}(0)) be a cut-off function with ϕ(y) = 1 in B_{1/2}^{2}(0). Let ψ(y′) ∈ C_{0}^{∞}(B_{1}^{2}(0)) be a cut-off function with ψ(y′) = 1 in B_{1/2}^{2}(0). We consider

f_{k}(a) = ∫_{B_{1}^{2}(0)×B_{1}^{2}(0)} | ▽ v_{k}|^{2}((x, x′) + a)ϕ(x)ψ(x′)dxdx′,

for a ∈ B_{2}^{2}(0)×B_{4}^{2}(0). Since v_{k} are harmonic maps, integration by parts leads to

\frac{∂f_{k}(a)}{∂a_{i}} = 2 ∑_{l=1}^{4} ∫_{B_{1}^{2}(0)×B_{1}^{2}(0)} \frac{∂v_{k}}{∂x_{l}} \frac{∂v_{k}}{∂x_{i}} ((x, x′) + a)\frac{∂(ϕψ)}{∂x_{l}}dxdx′

for i = 1, 2. By (15), we obtain

\frac{∂f_{k}(a)}{∂a_{i}} → 0 as k → ∞,

uniformly for a ∈ B_{2}^{2}(0) × B_{4}^{2}(0). It follows

∫_{B_{2}^{2}(0)×B_{1}^{2}(0)} | ▽ v_{k}|^{2}(x, x′ + b)dxdx′ ≤ \frac{ϵ_{0}}{8 · 2^{4}}

for each b ∈ B_{4}^{2}(0), and consequently we get

∫_{B_{2}^{2}(0)×B_{2}^{2}(0)} | ▽ v_{k}|^{2}(x, x′)dxdx′ ≤ ϵ_{0}.

Noting that v_{k} are harmonic maps, we see that v_{k} → v in

C^{2}(B_{1}^{2}(0) × B_{1}^{2}(0))

as k → ∞. By (14), v is a nonconstant quaternionic map. We choose x_{0} = (x_{0}^{1}, x_{0}^{2}, x_{0}^{3}, x_{0}^{4}) so that x_{0}^{3} + p^{3} ≠ 0, x_{0}^{4} + p^{4} ≠ 0, where p = (p^{1}, p^{2}, p^{3}, p^{4}) is the point in (13). We may assume that at x_{0} in ℝ^{4} and at v(x_{0}) in N the matrix expressions of the complex structures are given by (5). By (6), v satisfies

J^{3}v_{3} − J^{1}v_{4} = 0. (17)

Since u_{k} is a pseudo-tangent map of u, we have

∑_{i=1}^{4} x^{i} \frac{∂u_{k}}{∂x^{i}} = 0.

It follows that

∑_{i=1}^{4} (\frac{(x_{k}, x′_{k}) + r_{k}x_{0}}{r_{k}} )^{i} \frac{∂v_{k}}{∂x^{i}} (x_{0})

= ∑_{i=1}^{4} ((x_{k}, x′_{k}) + r_{k}x_{0}))^{i} \frac{∂u_{k}}{∂x^{i}} ((x_{k}, x′_{k}) + r_{k}x_{0})

= 0.

Since x_{0}^{3} + p^{3} ≠ 0, we have x_{0}^{3} + r_{k}^{−1}x_{k}^{′3} ≠ 0 for large k by (13) and then

\begin{aligned}\frac{∂v_{k}}{∂x^{3}} (x_{0}) = & − (\frac{x_{0}^{4} + r_{k}^{−1}x_{k}^{′4}}{x_{0}^{3} + r_{k}^{−1}x_{k}^{′3}} ) \frac{∂v_{k}}{∂x^{4}} (x_{0}) \\ & − ∑_{i=1}^{2} (\frac{x_{0}^{i} + r_{k}^{−1}x_{k}^{i}}{x_{0}^{3} + r_{k}^{−1}x_{k}^{′3}} ) \frac{∂v_{k}}{∂x^{i}} (x_{0}). \end{aligned} (18)

Since 0 ∈ Σ\sing(v) and |∇_{i}v| ≡ 0 for i = 1, 2,

lim_{k→∞} ∑_{i=1}^{2} (\frac{x_{0}^{i} + r_{k}^{−1}x_{k}^{i}}{x_{0}^{3} + r_{k}^{−1}x_{k}^{′3}} ) \frac{∂v_{k}}{∂x^{i}} (x_{0}) = 0.

Notice that

\frac{∂v}{∂x^{j}} (x_{0}) = lim_{k→∞} \frac{∂v_{k}}{∂x^{j}} (x_{0})

exist for j = 3, 4. Letting k → ∞ in (18), we have

\frac{∂v}{∂x^{3}} (x_{0}) = −\frac{x_{0}^{4} + p^{4}}{x_{0}^{3} + p^{3}} \frac{∂v}{∂x^{4}} (x_{0}). (19)

If the coefficient in (19) is infinite, then \frac{∂v}{∂x^{4}} (x_{0}) is zero and in turn \frac{∂v}{∂x^{3}} (x_{0}) is zero as well because v is quaternionic; if the coefficient is finite, then (19) means \frac{∂v}{∂x^{3}} (x_{0}) and \frac{∂v}{∂x^{4}} (x_{0}) are colinear, but by (17)

\frac{∂v}{∂x^{3}} (x_{0}) = −J^{3}J^{1} \frac{∂v}{∂x^{4}} (x_{0}) = −J^{2} \frac{∂v}{∂x^{4}} (x_{0}),

which implies \frac{∂v}{∂x^{3}} (x_{0}), \frac{∂v}{∂x^{4}} (x_{0}) are orthogonal, therefore we still conclude that

\frac{∂v}{∂x^{3}} (x_{0}) = \frac{∂v}{∂x^{4}} (x_{0}) = 0.

In sum, v ≡ constant since x_{0} is arbitrary as long as x_{0}^{3} + p^{3} ≠ 0, x_{0}^{4} + p^{4} ≠ 0. This contradicts to v is a nonconstant bubble, and consequently implies the compactness.

Step 2. We show that every pseudo-tangent map of u is smooth outside some rays.

Let ϕ be a pseudo-tangent map of u. It is a stationary quaternionic map of homogeneous degree zero in ℝ^{4}. We claim that ϕ|_{𝕊^{3}} is also stationary. Let X be a smooth vector field with compact support in ℝ^{4}, and let e_{1} = \frac{∂}{∂r} and e_{2}, e_{3}, e_{4} be an orthonormal frame on 𝕊^{3}. Assume that X = X^{i}e_{i} and X̃ = X|_{𝕊^{3}} = X̃^{2}e_{2} + X̃^{3}e_{3} + X̃^{4}e_{4}. We have

div(X) = \frac{∂X^{1}}{∂r} + \frac{3X^{1}}{r} + \frac{1}{r^{2}} div_{𝕊^{3}} X̃

and

X_{j}^{i} = \frac{X^{1}}{r} δ_{ij} + \frac{1}{r^{2}} X̃_{j}^{i} for i, j = 2, 3, 4.

Then

∫_{ℝ^{4}} (| ▽ ϕ|^{2}div(X) − 2⟨dϕ(▽_{i}X), dϕ(\frac{∂}{∂x^{i}})⟩) dV

= ∫_{ℝ^{4}} | ▽ ϕ|^{2} (\frac{∂X^{1}}{∂r} + \frac{X^{1}}{r} ) dV

+ ∫_{ℝ^{4}} \frac{1}{r^{4}} (| ▽_{𝕊^{3}} ϕ|^{2}div_{𝕊^{3}} X̃ − 2 ▽_{i}^{𝕊^{3}} ϕ ▽_{j}^{𝕊^{3}} ϕX̃_{j}^{i}) dV

= 0.

For any smooth vector field Ỹ on 𝕊^{3}, we choose X = η(r)Ỹ, where η ∈ C_{0}^{∞}(ℝ^{4}) is a cut-off function with η = 0 in (0, ϵ). Using the above identity, we obtain

∫_{𝕊^{3}} (| ▽_{𝕊^{3}} ϕ|^{2}div_{𝕊^{3}} Ỹ − 2 ▽_{i}^{𝕊^{3}} ϕ ▽_{j}^{𝕊^{3}} ϕỸ_{j}^{i}) dV = 0.

It follows that ϕ|_{𝕊^{3}} is stationary.

To prove ϕ is smooth outside some rays, it suffices to show that ψ = ϕ|_{𝕊^{3}} is smooth outside a finite set of points. The pseudo-tangent map of ψ is stationary, hence the 1-dimensional Hausdorff measure of its singular set is zero by Bethuel’s theorem in [2]. We then conclude that the pseudo-tangent maps of ψ are smooth except at 0, because otherwise the whole ray passing through a singular point different from the origin would be in the singular set.

We now derive the equation for the pseudo-tangent maps of ψ. For simplicity, we consider the pseudo-tangent map f of ψ at (1, 0, 0, 0). We choose a spherical coordinates in ℝ^{4} as follows:

x^{1} = r sin α sin β cos γ
x^{2} = r sin α sin β sin γ
x^{3} = r sin α cos β
x^{4} = r cos α.

(20)

In this coordinate system, (1, 0, 0, 0) = (1, π/2, π/2, 0). Suppose that J^{α} is expressed by J_{s}^{α} in the spherical coordinates. We can write J_{s}^{α} = A^{−1}J^{α}A for α = 1, 2, 3, where A is the Jacobi matrix of the change of coordinates (20). Then at (1, π/2, π/2, 0), we have

A = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) .

A simple computation shows that

A^{−1} = ( \begin{matrix}1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \end{matrix} ) , J_{s}^{1} = ( \begin{matrix}0 & 1 & 0 & 0 \\ −1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \end{matrix} ) ,

J_{s}^{2} = ( \begin{matrix}0 & 0 & 0 & −1 \\ 0 & 0 & −1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) , J_{s}^{3} = ( \begin{matrix}0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \end{matrix} ) .

We will use lower indices in f_{j}^{i} to denote the derivatives in ℝ^{4} at the blow-up point. Set

df = ( \begin{matrix}f_{1}^{1} & f_{2}^{1} & f_{3}^{1} & f_{4}^{1} \\ f_{1}^{2} & f_{2}^{2} & f_{3}^{2} & f_{4}^{2} \\ f_{1}^{3} & f_{2}^{3} & f_{3}^{3} & f_{4}^{3} \\ f_{1}^{4} & f_{2}^{4} & f_{3}^{4} & f_{4}^{4} \end{matrix} ) .

Then we have

J^{1} df J_{s}^{1} = ( \begin{matrix}f_{2}^{4} & −f_{1}^{4} & f_{4}^{4} & −f_{3}^{4} \\ −f_{2}^{3} & f_{1}^{3} & −f_{4}^{3} & f_{3}^{3} \\ f_{2}^{2} & −f_{1}^{2} & f_{4}^{2} & −f_{3}^{2} \\ −f_{2}^{1} & f_{1}^{1} & −f_{4}^{1} & f_{3}^{1} \end{matrix} )

J^{2} df J_{s}^{2} = ( \begin{matrix}−f_{4}^{2} & −f_{3}^{2} & f_{2}^{2} & f_{1}^{2} \\ f_{4}^{1} & f_{3}^{1} & −f_{2}^{1} & −f_{1}^{1} \\ f_{4}^{4} & f_{3}^{4} & −f_{2}^{4} & −f_{1}^{4} \\ −f_{4}^{3} & −f_{3}^{3} & f_{2}^{3} & f_{1}^{3} \end{matrix} )

J^{3} df J_{s}^{3} = ( \begin{matrix}f_{3}^{3} & −f_{4}^{3} & −f_{1}^{3} & f_{2}^{3} \\ f_{3}^{4} & −f_{4}^{4} & −f_{1}^{4} & f_{2}^{4} \\ −f_{3}^{1} & f_{4}^{1} & f_{1}^{1} & −f_{2}^{1} \\ −f_{3}^{2} & f_{4}^{2} & f_{1}^{2} & −f_{2}^{2} \end{matrix} ) .

Noting that f_{1} = 0 and f is a quaternionic map, we can see that f satisfies the equation

\{ \begin{matrix}f_{2}^{1} = −f_{3}^{2} − f_{4}^{3} \\ f_{2}^{2} = f_{3}^{1} − f_{4}^{4} \\ f_{2}^{3} = f_{4}^{1} + f_{3}^{4} \\ f_{2}^{4} = f_{4}^{2} − f_{3}^{3}. \end{matrix} (21)

When f is viewed as a map on ℝ^{3}, it is of homogeneous degree zero in ℝ^{3}. By an argument similar to the one used in the first step, we can show that, if a weakly convergence sequence of the pseudo-tangent maps of ψ did not converge strongly, we would get a nonconstant map bubble w satisfying the equations

0 = w_{3}^{2} + w_{4}^{3}
0 = w_{3}^{1} − w_{4}^{4}
0 = w_{4}^{1} + w_{3}^{4}
0 = w_{4}^{2} − w_{3}^{3}

and

x^{3}w_{3}^{i} + x^{4}w_{4}^{i} = 0 for i = 1, 2, 3, 4.

However, the above equations imply that w ≡ constant. This contradiction yields the compactness of the pseudo-tangent maps of ψ.

Step 3. Now the two requirements in Simon’s theorem are satisfied by ψ. Therefore ψ is smooth outside a finite set of points, and consequently ϕ is smooth outside the rays which pass through the singular points of ψ. Then the two requirements in Simon’s theorem are satisfied by u, so the singular set of u is a one dimensional rectifiable set. q.e.d.

Proposition 4.4. Let M be a compact hyperkähler surface and let N be a compact hyperkähler manifold with a real analytic metric which defines the hyperkähler structure. If N does not admit holomorphic 𝕊^{2} with respect to any complex structure in the hyperkähler 𝕊^{2} and u : M → N is a stationary quaternionic map, then it is smooth outside a finite set of points. Conversely, if N admits a holomorphic 𝕊^{2} with respect to some complex structure in the hyperkähler 𝕊^{2}, then there is a stationary quaternionic map u : ℝ^{4} → N whose singular set is of one dimension.

Proof. We continue the argument in the proof of Theorem 4.3, and we will provide in next section an alternative proof for the first part of the theorem.

We will show now that a nonconstant pseudo-tangent map f of ψ induces a holomorphic 𝕊^{2} in N. Note that f is of homogeneous degree zero in ℝ^{3}. Set h = f|_{𝕊^{2}}, we will show that h is a holomorphic 𝕊^{2} in N.

Using the spherical coordinates

x^{1} = r sin α cos θ
x^{2} = r sin α sin θ
x^{3} = r cos α,

by (21) and

\frac{∂}{∂x^{1}} = sin α cos θ \frac{∂}{∂r} + \frac{cos α cos θ}{r} \frac{∂}{∂α} − \frac{sin θ}{r sin α} \frac{∂}{∂θ}
\frac{∂}{∂x^{2}} = sin α sin θ \frac{∂}{∂r} + \frac{cos α sin θ}{r} \frac{∂}{∂α} + \frac{cos θ}{r sin α} \frac{∂}{∂θ}
\frac{∂}{∂x^{2}} = cos α\frac{∂}{∂r} − \frac{sin α}{r} \frac{∂}{∂α},

we can see that h satisfies the equation

\frac{1}{sin α} ( \frac{∂h^{1}}{∂θ} ; \frac{∂h^{2}}{∂θ} ; \frac{∂h^{3}}{∂θ} ; \frac{∂h^{4}}{∂θ} ) = ( \begin{matrix}0 & x_{3} & −x_{2} & −x_{1} \\ −x_{3} & 0 & x_{1} & −x_{2} \\ x_{2} & −x_{1} & 0 & −x_{3} \\ x_{1} & x_{2} & x_{3} & 0 \end{matrix} ) ( \frac{∂h^{1}}{∂α} ; \frac{∂h^{2}}{∂α} ; \frac{∂h^{3}}{∂α} ; \frac{∂h^{4}}{∂α} ) . (22)

One can check that h is a conformal harmonic map from 𝕊^{2} to N. In fact, h is holomorphic with respect to the the complex structures given by the 2 × 2-matrix with 0 on its diagonal and sin α and − sin^{−1} α as its (1, 2), (2, 1) entries respectively on 𝕊^{2} and the 4 × 4-matrix in (22) on N. This proves the first part of the proposition.

Conversely, if we have a holomorphic 𝕊^{2} in N given by h, then h satisfies the equation (22). We extend h to f : ℝ^{3} → N by letting f(r, α, θ) = h(α, θ) in spherical coordinates. Then in the standard coordinates, f satisfies the equation (21). We define u : ℝ^{4} → N by setting u(x^{1}, x^{2}, x^{3}, x^{4}) = f(x^{2}, x^{3}, x^{4}). One can check that u is a quaternionic map from ℝ^{4} to N with respect to the complex structures J_{s}^{α} at (1, π/2, π/2, 0) and J^{α} (α = 1, 2, 3). q.e.d.

By an argument similar to the one used in the proof of Theorem 4.3, we obtain the following well-known result for holomorphic maps.

Proposition 4.5. Let M be a compact complex surface and let N be a compact complex manifold with a real analytic Hermitian metric. Suppose that u ∈ W^{1,2}(M, N) is holomorphic outside a closed set Σ in M with H^{2}(Σ) < ∞. Then u can be extended to a holomorphic map from M possibly away from finitely many points to N.

Proof. It suffice to show that the pseudo-tangent maps of u are smooth except at 0. Suppose that ϕ is a pseudo-tangent map of u at a singular point, say (1, 0, 0, 0), let ψ = ϕ|_{𝕊^{3}}, we need only show that the pseudo-tangent maps of ψ are constant. Suppose that u is holomorphic with respect to J^{1} and J^{1}, then a pseudo-tangent map f of ψ satisfies the equation

f_{4}^{4} = f_{3}^{1}, − f_{3}^{4} = f_{4}^{1}, − f_{4}^{3} = f_{3}^{2}, f_{3}^{3} = f_{4}^{2}.

Since f also satisfies that x^{3}f_{3}+x^{4}f_{4} = 0, we can see that f ≡ constant. q.e.d.

5. Strong convergence

Let M compact hyperkähler surface and N be a compact hyperkähler manifold. Suppose that u_{k} is a sequence of quaternionic maps in a fixed homotopy class. We consider in this section when u_{k} will converge strongly in W^{1,2}(M, N).

Theorem 5.1. Let M be a compact hyperkähler surface and N be a compact hyperkähler manifold. Suppose that u_{k} is a sequence of quaternionic maps in a fixed homotopy class. If N does not admit holomorphic 𝕊^{2} with respect to the complex structure a_{i}J^{i} on ℝ^{4} restricts to 𝕊^{2} and the complex structure a_{i}J^{i} on N for some a_{i} (i = 1, 2, 3) with ∑_{i} a_{i}^{2} = 1, then a subsequence of {u_{k}} converges strongly in W^{1,2}(M, N) to a quaternionic map u. The singular set of the limit map u is a finite set of points.

The above theorem follows from the following theorem and the standard dimension reduction argument (cf. [8], [18], [20]).

Theorem 5.2. Let M compact hyperkähler surface and N be a compact hyperkähler manifold. Suppose that u_{k} is a sequence of quaternionic maps with bounded energies. If N does not admit holomorphic 𝕊^{2} with respect to the complex structure a_{i}J^{i} on ℝ^{4} restrict to 𝕊^{2} and the complex structure a_{i}J^{i} on N for some a_{i} (i = 1, 2, 3) with ∑_{i} a_{i}^{2} = 1, then there is a subsequence of {u_{k}} which converges strongly to a stationary quaternionic map u.

Proof. We can always assume that u_{k} ⇀ u weakly in W^{1,2}(M, N) and that | ▽ u_{k}|^{2}dx ⇀ | ▽ u|^{2}dx + ν in the sense of measure as k → ∞. Here ν is a nonnegative Radon measure on M with support in Σ, and Σ is the blow-up set of the sequence u_{k}. We will prove H^{2}(Σ) = 0. Otherwise, by an argument similar to the one used in the first step of the proof of Theorem 4.3, we get a nonconstant quaternionic map v with v_{i} = 0 for i = 1, 2.

Assume that e, a_{i}J^{i}e is an orthogonal basis of the normal bundle of T Σ at a point where T Σ exists, for some real valued functions a_{1}, a_{2}, a_{3} with a_{1}^{2}+a_{2}^{2}+a_{3}^{2} = 1. a_{i}’s may not be constant. Since v is a nonconstant harmonic map from 𝕊^{2} into N, viewed as extension of (T Σ)^{⊥}, v in fact is conformal. In particular,

⟨dv(e), dv(a_{j}J^{j}e)⟩ = 0
|dv(a_{j}J^{j}e)| = |dv(e)|.

If we set

dv(e) = ξ, (23)

then we may write

dv(a_{i}J^{i}e) = −b_{i}J^{i}ξ (24)

for some b_{i}, where b_{1}^{2} + b_{2}^{2} + b_{3}^{2} = 1. Note that dv restricts to 0 along Σ. We can check that

(a_{2}J^{1} − a_{1}J^{2})e ∈ Ker(dv) (25)

(a_{3}J^{1} − a_{1}J^{3})e ∈ Ker(dv). (26)

Solving the linear system (24), (25) and (26), we get

\{ \begin{matrix}dv(J^{1}e) & = & −a_{1}(b_{i}J^{i}ξ) \\ dv(J^{2}e) & = & −a_{2}(b_{i}J^{i}ξ) \\ dv(J^{3}e) & = & −a_{3}(b_{i}J^{i}ξ). \end{matrix} (27)

Since v is a quaternionic map, we have

∑_{α} J^{α}dvJ^{α}e = dv · e.

It follows from (27) that

ξ = ∑_{α} J^{α}dvJ^{α}e
= a_{1}b_{1}ξ − a_{1}b_{2}J^{3}ξ + a_{1}b_{3}J^{2}ξ + a_{2}b_{1}J^{3}ξ
+a_{2}b_{2}ξ − a_{2}b_{3}J^{1}ξ − a_{3}b_{1}J^{2}ξ + a_{3}b_{2}J^{1}ξ + a_{3}b_{3}ξ.

Notice that ξ, J^{1}ξ, J^{2}ξ, J^{3}ξ are linearly independent. Comparing coefficients in the above identity leads to

\{ \begin{matrix}a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} & = & 1 \\ a_{2}b_{3} − a_{3}b_{2} & = & 0 \\ a_{1}b_{3} − a_{3}b_{1} & = & 0 \\ a_{1}b_{2} − a_{2}b_{1} & = & 0. \end{matrix}

Solving this linear system, we have

a_{1} = b_{1} a_{2} = b_{2} a_{3} = b_{3}.

It follows that v is a holomorphic map from 𝕊^{2} to N with respect to the complex structure a_{i}J^{i} restrict to 𝕊^{2} and the complex structure a_{i}J^{i} on N. But no such holomorphic can exist by assumption. So we must have H^{2}(Σ) = 0 and in turn u_{k} converge strongly to u in W^{1,2} norm. q.e.d.

References

[1] W. Allard, First variation of a varifold, Ann. of Math. 95 (1972) 419–491.

[2] F. Bethuel, On the singular set of stationary harmonic maps, Manu. Math. 78 (1993) 417–443.

[3] J. Chen, Complex anti-self-dual connections on product of Calabi-Yau surfaces and triholomorphic curves, Commun. Math. Phys. 201 (1999) 201–247.

[4] J. Chen & G. Tian, Minimal surfaces in Riemannian 4-manifolds, GAFA, 7 (1997) 873–916.

[5] S. S. Chern & J. Wolfson, Minimal surfaces by moving frames, Ann. of Math. 105 (1983) 59–83.

[6] S. Donaldson & R. Thomas, Gauge theory in higher dimensions, The Geometric Universe: Science, Geometry and the work of Roger Penrose, (S.A. Huggett et al eds.), Oxford Univ. Press, Oxford, 1998.

[7] L. C. Evans, Partial regularity for stationary harmonic maps, Arch. Rat. Mech. and Anal. 116 (1991) 101–112.

[8] H. Federer, Geometric measure theory, Springer, 1969.

[9] J. M. Figueroa-O’Farrill, C. Köhl & B. Spence, Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves, Nuclear Phys. B 521 (1998) 419–443.

[10] R. Fueter, Die Funktionentheorie der Differentialgleichungen Δu = 0 and ΔΔu = 0 mit vier reellen Variablen, Comment. Math. Helv. 7 (1935) 307–330.

[11] D. Gilbarg & N. Trudinger, Elliptic partial differential equations of second order, Springer, 1983.

[12] F. Hélein, Regularite des applications faiblement harmoniques entre-une surface et une sphere, CRAS 311 (1990) 519–524.

[13] N. Hitchin, Hyper-Kähler manifolds, Séminaire Bourbaki, Vol. 1991/92. Astérisque No. 206 (1992), Exp. No. 748, 3, 137–166.

[14] M. Itoh, Quaternion structure on the moduli space of Yang-Mills connections, Math. Ann. 276 (1987) 581–593.

[15] D. Joyce, Hypercomplex algebraic geometry, Quart. J. Math. Oxford 49 (1998) 129–162.

[16] A. Lichnerowicz, Applications harmoniques et variétés kählériennes, Symp. Math. III, Bologna, 1970, 341–402.

[17] J. Li & G. Tian, A blow-up formula for stationary harmonic maps, IMRN 14 (1998) 735–755.

[18] F.-H. Lin, Gradient estimates and blow-up analysis for stationary harmonic maps. I, Ann. of Math. 149 (1999) 785–829.

[19] S. Mukai, Symplectic structure of the moduli space of stable sheaves on an abelian or K3 surface, Invent. Math. 77 (1984) 101–116.

[20] R. Schoen & K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom. 17 (1982) 307–336.

[21] J. Sacks & K. Uhlenbeck, The existence of minimal immersions of two spheres, Ann. of Math. 113 (1981) 1–24.

[22] L. Simon, Rectifiability of the singular set of energy minimizing maps, Calc. Var. 3 (1995) 1–65.

[23] , Lectures on geometric measure theory, Proc. Center Math. Anal. Vol. 3, 1983, Australian National Univ. Press.

[24] E. Stein, Singular integral and differential property of functions, Princeton Univ. Press, Princeton, 1970.

[25] A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Phil. Soc. 85 (1979) 199–225.

[26] G. Tian, Yang-Mills equations and calibrated geometry. I, Ann. of Math. 151 (2000) 193–268.

[27] J. Wolfson, Minimal surfaces in Kähler surfaces and Ricci curvature, J. Differential Geom. 29 (1989) 281–294.

The University of British Columbia, Canada Academia Sinica, Beijing, China