2 Existence of an energy minimizer

In this section we prove the existence of an energy-minimizing biharmonic almost complex structure on $(M, g)$ and derive the Euler-Lagrange equation. We recall the definition of the Sobolev spaces of almost complex structures.

Definition 2.1. Given an almost Hermitian manifold $(M, g)$ with compatible almost complex structures in ${\mathcal J}_g$ , we define $W^{k, p}({\mathcal J}_g)$ to be the subspace of $W^{k, p}(T^*M\otimes TM)$ consisting of those sections $J\in W^{k, p}(T^*M\otimes TM)$ , satisfying the compatible condition almost everywhere,

(2.1) $$ \begin{align} J^2=-id, g(J\cdot, \cdot)+g(\cdot, J\cdot)=0. \end{align} $$

We have the following.

Proof. We only prove the existence of $W^{2, 2}$ energy minimizer of ${\mathcal E}_2(J)$ over $W^{2, 2}({\mathcal J}_g).$ We will derive the Euler-Lagrange equation later. This is a standard practice of calculus of variations. Take an energy-minimizing sequence $J_k\in W^{2, 2}({\mathcal J}_g)$ , such that

$$\begin{align*}{\mathcal E}_2(J_k)\rightarrow \inf_{J\in W^{2, 2}({\mathcal J}_g)} {\mathcal E}_2(J). \end{align*}$$

Since $J_k\in W^{2, 2}(TM\otimes T^*M)$ , it follows from Kondrachov compactness that a subsequence, still denoted by $J_k$ , converges strongly in $W^{1, 2}(T^*M\otimes TM)$ and weakly in $W^{2, 2}(T^*M\otimes TM)$ . Denote the limit by $J_0\in W^{2, 2}(TM\otimes T^*M)$ . The strong convergence in $W^{1, 2}$ implies, in particular, that $J_k$ converges to $J_0$ almost everywhere; therefore, $J_0$ satisfies the compatible condition (2.1) almost everywhere. Hence, $J_0\in W^{2, 2}({\mathcal J}_g)$ . The weak convergence in $W^{2, 2}$ implies that

$$\begin{align*}{\mathcal E}_2(J_0)\leq \liminf {\mathcal E}_2(J_k)=\inf_{J\in W^{2, 2}({\mathcal J}_g)} {\mathcal E}_2(J). \end{align*}$$

This forces that ${\mathcal E}_2(J_0)=\min {\mathcal E}_2(J)$ , and $J_0$ is an energy minimizer. Moreover, this also implies that the convergence $J_k\rightarrow J_0$ is strongly in $W^{2, 2}$ and that the set of all energy minimizers is compact.

Straightforward computation gives the Euler-Lagrange equation in smooth setting

(2.2) $$ \begin{align} \Delta^2 J+J(\Delta^2 J) J=0. \end{align} $$

An equivalent form is $[\Delta ^2 J, J]=0$ . By $J^2=-id$ , we can rewrite the equation as,

(2.3) $$ \begin{align} \Delta^2 J=Q(J, \nabla J, \nabla^2 J, \nabla^3 J), \end{align} $$

where Q is given by

(2.4) $$ \begin{align} Q=J\Delta J \Delta J+J\nabla_pJ\nabla_p\Delta J+J\nabla_p\Delta J\nabla_pJ+J\Delta(\nabla_pJ\nabla_pJ). \end{align} $$

We need to be more careful when working in $W^{2, 2}({\mathcal J}_g)$ to derive equations for weak solutions. For any $A\in TM\rightarrow TM$ , it is convenient to define its adjoint $A^*$ as follows.

$$\begin{align*}g(A^*\cdot, \cdot )=g(\cdot, A\cdot). \end{align*}$$

Hence, g-compatibility can be written as $A+A^*=0$ . For any $J\in {\mathcal J}_g$ , the tangent space ${\mathcal S}_J$ of J can be described as the endomorphisms $S: TM\rightarrow TM$ as

$$\begin{align*}{\mathcal S}_J=\{S: SJ+JS=0, S+S^*=0\}. \end{align*}$$

Clearly, $S\in {\mathcal S}_J$ , and then $JS$ is in ${\mathcal S}_J$ as well. The adjoint can be directly carried over for Sobolev spaces. Necessary properties of the adjoint endomorphism are summarized in [Reference He and Jiang6, Proposition 4.1]. For any $S\in {\mathcal S}_J\cap L^\infty $ , it is direct to check that $J(t)=J \exp (tSJ)$ gives a path in ${\mathcal J}_g\cap L^\infty $ for $|t|<<1$ , with $\partial _t J(0)=S$ .

Proposition 2.3. A critical point of ${\mathcal E}_2(J)$ on $W^{2, 2}({\mathcal J}_g)$ satisfies the Euler-Lagrange equation (2.3) in the following weak sense. For any $T\in W^{2, 2}(T^*M\otimes TM)\cap L^\infty $ , we have

(2.5) $$ \begin{align} \int_M (\Delta J-J\nabla J\nabla J, \Delta T)dv+\int_M (A, T)dv+\int_M(B, \nabla T) dv=0, \end{align} $$

where we write

$$ \begin{align*} A=&J\Delta J\Delta J+\nabla_p J\nabla_p J\Delta J-\Delta J\nabla_p J\nabla_p J +\nabla_p J\Delta J\nabla_p J \nonumber \\ B=&\nabla J \Delta J J+J\Delta J\nabla J. \end{align*} $$

Proof. Assume that $T\in W^{2, 2}(\Gamma (T^*M\otimes TM))\cap L^\infty $ . Denote $R=T+JTJ$ and $S=R-R^*$ . Then S satisfies $SJ+JS=0, S+S^*=0$ . Taking variation in the form $J(t)=J\exp (tSJ)$ for small $|t|<<1$ , we have $\int _M (\Delta J, \Delta S)dv=0.$ Since J is g-compatible, this implies $\int _M (\Delta J, \Delta R)dv=0$ . In other words, we have

$$\begin{align*}\int_M (\Delta J, \Delta T)dv+\int_M (\Delta J, \Delta (JTJ))dv=0. \end{align*}$$

For $J\in W^{2, 2}$ , we compute

$$\begin{align*}\Delta (JTJ)=J (\Delta T) J+\Delta J T J+J T\Delta J+2\nabla J \nabla T J+2\nabla J T\nabla J+2J\nabla T\nabla J. \end{align*}$$

Since J is g-compatible, it is straightforward to check that

$$ \begin{align*} (\Delta J, J \Delta T J)=&(J\Delta J J, \Delta T)\nonumber \\ (\Delta J, \Delta J T J)=&(\Delta J \Delta J J, T). \end{align*} $$

Hence, we get that

(2.6) $$ \begin{align} \int_M (\Delta J+J\Delta J J, \Delta T)+ (A_0, T)+2(B, \nabla T)=0, \end{align} $$

where we have

$$ \begin{align*} A_0=&\Delta J\Delta J J+J\Delta J\Delta J +2\nabla_p J\Delta J\nabla_p J, \nonumber \\ B=&\nabla J \Delta J J+J\Delta J\nabla J. \end{align*} $$

However, we use $\Delta (J^2)=0$ to conclude that

$$ \begin{align*} &\Delta J J+J\Delta J+2\nabla_p J\nabla_p J=0 \nonumber \\ &J\Delta J J=\Delta J-2J\nabla_p J\nabla_p J. \end{align*} $$

We compute that

$$\begin{align*}\Delta J\Delta J J=J\Delta J\Delta J+2\nabla_p J\nabla_p J\Delta J-2\Delta J\nabla_p J\nabla_p J. \end{align*}$$

Hence, we get that

$$\begin{align*}A_0=2\left(J\Delta J\Delta J+\nabla_p J\nabla_p J\Delta J-\Delta J\nabla_p J\nabla_p J +\nabla_pJ\Delta J\nabla_p J\right). \end{align*}$$

We denote $A_0=2A$ , with

$$\begin{align*}A=J\Delta J\Delta J+\nabla_p J\nabla_p J\Delta J-\Delta J\nabla_p J\nabla_p J +\nabla_p J\Delta J\nabla_p J. \end{align*}$$

Together with (2.6), we have

(2.7) $$ \begin{align} \int_M \left(\Delta J-J\nabla J\nabla J, \Delta T\right)+(A, T)+(B, \nabla T)=0. \end{align} $$

This completes the proof.

Proposition 2.3 is equivalent to (2.2) for $J\in W^{2, 2}$ when (2.2) is understood as a weak solution in the form that

(2.8) $$ \begin{align} \int_M (\Delta J, \Delta T)dv+\int_M (\Delta J, \Delta(JTJ) ) dv=0. \end{align} $$

An almost complex structure $J\in W^{2, 2}({\mathcal J}_g)$ is called a weak biharmonic almost complex structure if it satisfies (2.8) or its equivalent form (2.5). The equivalent descriptions below will be useful as well.

Proposition 2.4. A weak biharmonic almost complex satisfies the following.

(2.9) $$ \begin{align} \int_M (\Delta J, \Delta (TJ-JT))dv=0, \end{align} $$

where $T\in \Gamma (TM\otimes T^*M)$ . Or equivalently, we have the following.

(2.10) $$ \begin{align} \int_M (\Delta J, \Delta T J-J\Delta T)dv+2\int_M (\Delta J, \nabla T\nabla J-\nabla J\nabla T)dv=0. \end{align} $$

As a consequence, the weak limit of a sequence of biharmonic almost complex structure with bounded $W^{2, 2}$ norm is biharmonic.

Proof. To derive (2.9), we take $T=\tilde T J$ in (2.8) and observe that

(2.11) $$ \begin{align} \int_M (\Delta J, (\Delta J) T)dv=\int_M (\Delta J, T \Delta J)dv. \end{align} $$

Similarly, we have

$$\begin{align*}(\Delta J, T\Delta J)=-(\Delta J\Delta J, T). \end{align*}$$

This gives (2.10). Suppose a sequence of weak biharmonic almost complex structures $J_k$ , with bounded $W^{2, 2}$ norm, converges weakly in $W^{2, 2}$ to $J_0$ and strongly in $W^{1, 2}$ . By passing to the limit, $J_0$ satisfies (2.10). More precisely, for any fixed smooth T, we have

$$\begin{align*}\int_M (\Delta (J_k-J_0), \Delta T J_0-J_0\Delta T)dv \rightarrow 0. \end{align*}$$

The strong convergence $J_k\rightarrow J_0$ in $L^2$ implies that

$$\begin{align*}\int_M (\Delta J_k, \Delta T (J_k-J_0)-(J_k-J_0)\Delta T)dv\rightarrow 0. \end{align*}$$

Together, this implies that

$$\begin{align*}\int_M (\Delta J_k, (\Delta T)J_k-J_k\Delta T)dv\rightarrow \int_M (\Delta J_0, (\Delta T)J_0-J_0\Delta T)dv. \end{align*}$$

Similarly, we have

$$\begin{align*}\int_M (\Delta J_k, \nabla T\nabla J_k-\nabla J_k\nabla T)dv\rightarrow \int_M (\Delta J_0, \nabla T\nabla J_0-\nabla J_0\nabla T)dv. \end{align*}$$

This completes the proof.

He-Jiang [Reference He and Jiang6, Theorem 2, Corollary 1] proved a smooth regularity for a general elliptic system. As a result, we have the following.

3 Biharmonic almost complex structures in a homotopy class

A fundamental problem in the theory of harmonic maps is finding harmonic maps in a fixed homotopy class. Similarly, we would like to ask the same question for biharmonic almost complex structures. We shall see that the topology of M plays a very important role. The following famous example about almost complex structures on a $K3$ surface constructed by Donaldson [Reference Donaldson3] serves as an important example. We recall relevant discussions. A compatible almost complex structure J on an oriented Riemannian four-manifold M can be considered as a section of the associated $SO(4)/U(2)$ -bundle over M (the sphere bundle of $\Lambda ^2_+$ , known as the ‘twistor space’). Let A denote the set of homotopy classes of almost complex structures, so the first Chern class gives a map

$$\begin{align*}c_1: A\rightarrow H^2(M, \mathbb{Z}). \end{align*}$$

Donaldson [Reference Donaldson3, Section 6] defined a map $p: A\rightarrow A$ with $p^2=\text {id}$ and $c_1\circ p=c_1$ as follows. For $\sigma \in A$ , $p(\sigma )$ agrees with $\sigma $ outside a small ball in M, and over this ball, the two compare by the nonzero element of

$$\begin{align*}[S^4, SO(4)/U(2)]=[S^4, S^2]\cong \mathbb{Z}/2. \end{align*}$$

We need the following general result about $\sigma $ and $p(\sigma )\in A$ , which we have learned from Teichner through mathoverflow [Reference Teichner11].

The main result in this section is to prove Theorem 1.3. The proof consists of several steps. Let $J_k\in \sigma $ be a minimizing sequence such that ${\mathcal E}_2(J_k)\rightarrow E_0$ . A direct computation shows that $J_k$ has uniformly bounded $W^{2, 2}$ norm. Hence, we can assume that $J_k$ converges to $J_0$ weakly in $W^{2, 2}$ and strongly in $W^{1, 2}$ , in particular $J_0\in W^{2, 2}({\mathcal J}_g)$ . First, we have the following.

Proof. We only need to prove that $J_0$ is a weak biharmonic almost complex structure satisfying (2.10). Let $J_k$ be a minimizing sequence and $J_0$ be its weak limit in $W^{2, 2}({\mathcal J}_g)$ . For any fixed smooth ${T\in \Gamma (TM\otimes T^*M)}$ , consider

$$\begin{align*}S_t= t (T+J_k TJ_k)-t(T+J_kTJ_k)^{*}. \end{align*}$$

Note that for $|t|$ sufficiently small, $S_t$ has small $L^\infty $ norm and we construct

$$\begin{align*}J_k(t)=J_k \exp(S_t)\in {\mathcal J}_g. \end{align*}$$

We compute, for $|t|$ sufficiently small,

(3.1) $$ \begin{align} E(J_k^t)=\int_M |\Delta J_k^t|^2 dv=E(J_k)+4t\int_M (\Delta J_k, \Delta (J_kT-TJ_k))dv+O(t^2), \end{align} $$

where the term $O(t^2)$ denotes the terms of higher order in t. We have $|O(t^2)|\leq Ct^2$ for a uniformly bounded constant C (assuming t is small). A quick way to see (3.1) is to write the (matrix) expansion of $J_k(t)$ as

$$ \begin{align*} J_k(t)&=J_k+t(J_kT-TJ_k)-t(J_kT-TJ_k)^*+O(t^2). \end{align*} $$

Since $J_k$ is a minimizing sequence, hence when $k\rightarrow \infty $ , we have

$$\begin{align*}\liminf_{k\rightarrow \infty} E(J_k^t)-E(J_k) \geq 0. \end{align*}$$

Hence, for $|t|$ sufficiently small,

$$\begin{align*}\limsup_k 4t\int_M (\Delta J_k, \Delta (J_kT-TJ_k))dv\geq \liminf_k 4t\int_M (\Delta J_k, \Delta (J_kT-TJ_k))dv\geq 0. \end{align*}$$

In particular, this implies that

$$\begin{align*}\lim\sup_k(\Delta J_k, \Delta (J_kT-TJ_k))dv=\liminf_k \int_M (\Delta J_k, \Delta (J_kT-TJ_k))dv=0. \end{align*}$$

Similar as in (2.10) and (2.11), we use the fact

$$\begin{align*}\int_M (\Delta J_k, \Delta J_k T)dv=\int_M(\Delta J_k, T\Delta J_k)dv \end{align*}$$

to conclude that

$$\begin{align*}\lim_k \int_M (\Delta J_k, J_k\Delta T-\Delta T J_k)dv+2\int_M (\Delta J_k, \nabla J_k \nabla T-\nabla T\nabla J_k)dv=0. \end{align*}$$

Since $J_k$ converges to $J_0$ weakly in $W^{2, 2}$ and strongly in $W^{1, 2}$ , the above implies that

$$\begin{align*}\int_M (\Delta J_0, J_0\Delta T-\Delta T J_0)dv+2\int_M (\Delta J_0, \nabla J_0 \nabla T-\nabla T\nabla J_0)dv=0. \end{align*}$$

By Proposition 2.4, $J_0$ is a weak biharmonic almost complex structure. Hence it is smooth by regularity results in [Reference He and Jiang6].

An important question is to understand the homotopy class of $J_0$ . To proceed, we need the following fact about the first Chern class.

Lemma 3.3 (Wood [Reference Wood16]).

Given a compact almost Hermitian manifold $(M, g, J, \omega )$ , then the first Chern class can be represented by the following 2-form $\gamma $ such that

(3.2) $$ \begin{align} 2\pi \gamma={\mathcal R}(\omega)+\chi,\; \text{with}\; \chi(X, Y):=\frac{1}{4}\omega(\nabla _X J, \nabla_Y J), \end{align} $$

where ${\mathcal R}(\omega )$ is the curvature operator acting on the Kähler form $\omega $ .

Lemma 3.3 has the following important consequence.

Proof. The first Chern class $c_1(J)$ is the deformation invariant, and hence it remains the same in the homotopy class $\sigma $ . By Lemma 3.3, the first Chern class is represented by the 2-form $\gamma $ . We write $\gamma _k$ for $J_k$ and $\gamma _0$ for $J_0$ . Since $J_k$ converges strongly to $J_0$ in $W^{1, 2}$ , then for any smooth two form $\zeta $ , (3.2) implies that

$$\begin{align*}\lim_{k\rightarrow \infty}\int_M \gamma_k\wedge \zeta=\int_M \gamma_0\wedge \zeta. \end{align*}$$

It follows that $[\gamma _k]=[\gamma _0]\in H^2(M, \mathbb {Z})$ .

Hence, the fist Chern class remains unchanged for the weak limit. If M is simply connected and nonspin, Lemma 3.1 implies that $J_0$ is in the homotopy class $\sigma $ .

If M is simply connected and spin, the situation is more complicated. By Lemma 3.1, $J_0$ is either in the class $\sigma $ or in $p(\sigma )$ . In the former case, $J_0$ is an energy-minimizing biharmonic almost complex structure in $\sigma $ ; in the later case, we prove that there is an energy-minimizing biharmonic almost complex structure in $p(\sigma )$ . To achieve this, we consider the homotopy class $\sigma $ and $p(\sigma )$ simultaneously. We consider $\inf _{J\in \sigma } {\mathcal E}_2(J)$ and $\inf _{J\in p(\sigma )}{\mathcal E}_2(J)$ . We assume that $\inf _{J\in \sigma } {\mathcal E}_2(J)\geq \inf _{J\in p(\sigma )}{\mathcal E}_2(J)$ . A minimizing sequence $J_k$ in $p(\sigma )$ will have a weak limit $J_0$ and ${\mathcal E}_2(J_0)\geq \inf _{J\in p(\sigma )}{\mathcal E}_2(J)$ , since $J_0$ is either in $\sigma $ or $p(\sigma )$ . This will force that $J_0$ is an energy-minimizing biharmonic almost complex structure.

When M is not necessarily simply connected, Lemma 3.1 does not hold anymore. We need more precise control of weak convergence to obtain the following.

The key is to prove a version of $\epsilon $ -regularity for the minimizing sequence $J_k$ . As a consequence, $J_k$ converges strongly in $W^{2, 2}$ to $J_0$ except around finitely many isolated points. This will imply that the homotopy class of $J_0$ is either $\sigma $ or $p(\sigma )$ .

Now we are ready to state and prove the $\epsilon $ -regularity for the minimizing sequence $J_k$ (see Lemma 3.8 and Lemma 3.9). Fix a sufficiently small positive number $\epsilon _0$ , which depends only on $(M, g)$ and will be specified later. Let $\iota $ be the injectivity radius of $(M, g)$ . Suppose a minimizing sequence $J_k\in \sigma $ converges weakly to $J_0$ in $W^{2, 2}$ and strongly in $W^{1, 2}$ . For $r\in (0, \iota )$ , $p\in M$ and $J\in W^{2, 2}$ , denote

(3.3) $$ \begin{align} &E(r, p)=\int_{B_r(p)}|\Delta J|^2 dv \nonumber \\ &F(r, p)=\int_{B_r(p)} (|\nabla^2J|^2+|\nabla J|^4)dv. \end{align} $$

We write $E_0(r, p), F_0(r, p), E_k(r, p), F_k(r, p)$ correspondingly for $J_0$ and $J_k$ . Set ${\mathcal S}_r=\{p\in M: \liminf _{k\rightarrow \infty } F_k(r, p)\geq \epsilon _0\}.$ Clearly, ${\mathcal S}_r\subset {\mathcal S}_s$ for $r<s$ . Denote

$$\begin{align*}{\mathcal S}:=\cap_{r>0} {\mathcal S}_r=\lim_{r\rightarrow 0}{\mathcal S}_r.\end{align*}$$

We introduce the measures which are all totally bounded,

(3.4) $$ \begin{align}\begin{cases} & \!\!\!\!\!d\mu_k=(|\nabla^2J_k|^2+|\nabla J_k|^4)dv\\ &\!\!\!\!\! d\mu_0=(|\nabla^2 J_0|^2+|\nabla J_0|^4)dv\\ & \!\!\!\!\! d\xi_k=|\Delta J_k|^2dv\\ &\!\!\!\!\! d\xi_0=|\Delta J_0|^2dv. \end{cases}\end{align} $$

By passing to a subsequence, $\mu _k$ converges weakly to a positive Radon measure $\mu $ , and $\xi _k$ converges weakly to a positive Radon measure $\xi $ . By Fatou’s lemma, there exist positive Radon measures $\nu $ and $\lambda $ (called the defect measure [Reference Lin8, Reference Lin9]), such that

$$\begin{align*}\begin{cases} &\!\!\!\!\! d\mu=d\nu+d\mu_0\\ &\!\!\!\!\! d\xi=d\lambda+d\xi_0. \end{cases}\end{align*}$$

Certainly, $J_k$ converges to $J_0$ strongly in $W^{2, 2}$ if and only if either $\nu \equiv 0$ or $\lambda \equiv 0$ . But $\nu $ and $\lambda $ are not necessarily the same in general. The interplay between two defect measures $\nu $ and $\lambda $ makes our discussions below more complicated than the theory of the harmonic maps, where only the measure $|\nabla u|^2dv$ comes to play. We have the following.

Proof. First, it is straightforward to see that ${\mathcal S}$ is contained in the support of $\nu $ . Since $J_0$ is smooth, we have for any $x\in M$ , $\lim _{r\rightarrow 0}\mu _0(B_r(x))=0.$ If $x\in {\mathcal S}$ , then

$$\begin{align*}\lim_{r\rightarrow 0}\nu(B_r(x))=\lim_{r\rightarrow 0}[\mu(B_r(x))-\mu_0(B_r(x))]\geq \epsilon_0. \end{align*}$$

This shows that ${\mathcal S}$ is contained in the support of $\nu $ .

Next, we claim the following. For $r\in (0, r_0]$ , where $r_0$ is a fixed, sufficiently small number, if $\mu (B_{2r}(p))<\epsilon _0$ , then $\nu \equiv 0$ in $B_{r/2}(p)$ .

We sketch the idea of the proof briefly. If $\mu (B_{2r}(p))<\epsilon _0$ for $\epsilon _0$ sufficiently small, then after passing to a subsequence, we have

$$\begin{align*}\int_{B_{2r}(p)} (|\nabla^2 J_k|^2+|\nabla J_k|^4)dv<2\epsilon_0 \end{align*}$$

for sufficiently large k. This implies that $J_k$ is close to its average in $B_{2r}$ and the same discussion holds for $J_0$ . In particular, $J_k$ and $J_0$ , when restricted in $B_r$ , are homotopy to each other. The key is to construct a new ‘almost minimizing’ sequence $\tilde J_k$ in the same homotopy class such that $\tilde J_k=J_0$ inside $B_{r/2}(p)$ and $\tilde J_k=J_k$ outside $B_r(p)$ , while the behavior in the annulus region is precisely controlled. This would imply $\nu , \lambda \equiv 0$ in $B_{r/2}(p)$ .

Suppose at the moment, the claim is established. If p is in the support of $\nu $ , then $\mu (B_{2r}(p))\geq \epsilon _0$ for all sufficiently small r. Hence $p\in {\mathcal S}$ . Since the total energy is bounded, it follows that ${\mathcal S}$ contains, at most, finitely many isolated points. We complete the proof given Lemma 3.9 below, where we establish the claim.

First, we specify the choice of $r_0$ . We can do the scaling $g_r=r^{-2}g$ for $r\leq r_0$ . We choose $r_0$ sufficiently small, such that $g_r$ is sufficiently close to the Euclidean metric in the ball $B_2(p)$ (we identify $B_2(p)$ with the Euclidean ball $B_2$ equipped with the metric $g_r$ ) such that

(3.5) $$ \begin{align} |g_{ij}-\delta_{ij}|+\sum_{k=1}^4|D^k g_{ij}|<\delta_0,\end{align} $$

where $\delta _0$ measures how close the metric $g_r$ is with respect to the Euclidean metric in $B_2$ . We also assume that $g_{ij}(0)=\delta _{ij}, \partial g_{ij}(0)=0$ .

Note that we choose $r_0, \delta _0$ and $\epsilon _0$ uniformly for any point $p\in M$ . The constants $r_0, \delta _0$ and $\epsilon _0$ are all fixed. Since r will also be fixed and the energy functionals are scaling invariant, we consider $(M, g_r)$ instead of $(M, g)$ . In other words, we can assume that, by scaling if necessary, g satisfies (3.5) in any geodesic ball $B_2(p)\subset M$ such that the injectivity radius of $(M, g)$ is bigger than $2$ . We identify $(B_2(p), g)$ with the Euclidean ball $(B_2, g)$ via the exponential map $\exp _p: T_pM\rightarrow M$ . Over $(B_2, g)$ , the tangent bundle is trivial and we write $J: TM\rightarrow TM$ over $(B_2, g)$ as a matrix-value function $J(x)$ . We use $\nabla , \nabla ^2$ , etc. to denote covariant derivatives of with respect to g over $B_2$ . We will also use $D, D^2$ to denote the Euclidean derivatives over $B_2$ . After this choice of the scaling and local coordinates, an almost complex structure J over $(B_2, g)$ is a matrix-valued function, which we still denote as J. We establish the main technical lemma.

The proof of this lemma involves a construction of a sequence of ‘almost energy-minimizing’ almost complex structures $\tilde J_k$ in $\sigma $ such that

(3.6) $$ \begin{align} \tilde J_k(x)=\begin{cases} &\!\!\!\! J_k(x), x\in M\setminus B_1(p)\\ &\!\!\!\! J_0(x), x\in B_{1-j^{-1}}(p), \end{cases} \end{align} $$

where $k\geq k_j=k(j)$ is sufficiently large depending on j. In the end, we will let $j\rightarrow \infty $ (and $k_j\rightarrow \infty $ accordingly) to get an almost energy-minimizing subsequence. Such a construction is a type of extension of an almost complex structure which equals $J_k$ in $M\setminus B_1(p)$ and which equals $J_0$ in $B_{1-j^{-1}}(p)$ . The construction happens in a small annulus $B_1(p)\setminus B_{1-j^{-1}}(p)$ . The small energy condition $\mu (B_{2}(p))\leq \epsilon _0$ plays a very important role. In particular, this implies that $\tilde J_k$ is still in $\sigma $ , using a theorem of White [Reference White15].

The construction involves several delicate choices of small constants and cutoff functions. We shall first briefly explain the process of construction, leaving details to be proved below. We work on $(B_2, g)$ . Let $\psi _j: [0, \infty )\rightarrow [0, 1]$ be a smooth cutoff function depending on j such that

$$\begin{align*}\psi_j(s)= \begin{cases} &\!\!\!\! 1, s\geq 1\\ &\!\!\!\! 0, s\leq 1-j^{-1} \end{cases} \end{align*}$$

with bounds $|\psi ^{'}_j|\leq 3j, |\psi ^{"}_j|\leq 10 j^2$ . For $x\in B_2$ , we denote

$$\begin{align*}J_{k,\, j}(x)=J_k(x)+(J_0(x)-J_k(x))(1-\psi_j(|x|)). \end{align*}$$

Note that

(3.7) $$ \begin{align} J_{k,\, j}(x)=\begin{cases}& \!\!\!\!\! J_k(x), |x|\geq 1\\ &\!\!\!\!\! J_0(x), |x|\leq 1-j^{-1}. \end{cases} \end{align} $$

We extend $J_{k,\, j}$ to M such that it equals $J_k$ outside $B_1(p)$ . Note that $J_{k,\, j}$ might not even be invertible for some points $|x|\in (1-j^{-1}, 1)$ since the convergence of $J_k\rightarrow J_0$ does not imply the convergence in $L^\infty $ . To overcome this difficulty, we construct a smooth approximation of $J_{k,\, j}$ using a local average technique (a modification of mollifier).

Let $\phi (x)=\phi (|x|)$ be a nonnegative smooth radial cutoff function which is supported in $B=B_1$ with $\int _B \phi dx=1$ . For any given J and $\rho>0$ , we denote

(3.8) $$ \begin{align} J_\rho(x)=\int_{B_\rho (x)}\phi_\rho(y-x) J(y)dy=\int_{B} \phi(z)J(x+\rho z)dz, \end{align} $$

where we use the notation, $\phi _\rho (x)=\rho ^{-4} \phi \left (\frac {x}{\rho }\right ). $ When $\rho =0$ , $\phi _\rho $ is the delta-function and $J_\rho (x)=J(x)$ (this is also clear from the second equality in (3.8)). Certainly, $J_\rho $ is the smooth approximation of J, and $J_\rho $ converges to J (in a certain norm depending on the regularity of J) when $\rho \rightarrow 0$ . However, such a smooth approximation does not preserve (3.7) in general. Hence, we allow $\rho $ to be dependent of $|x|$ and we write $\rho : [0, 2]\rightarrow [0, 1]$ . Such a technique is a modification of Schoen-Uhlenbeck [Reference Sacks and Uhlenbeck12]. The support of $\rho $ is contained in $(1-j^{-1}, 1)$ . In other words,

(3.9) $$ \begin{align} \rho(s)=0, s \in [0, 1-j^{-1}]\cup [1, 2]. \end{align} $$

We choose $\rho (1-(2j)^{-1})=\max \rho =\bar \rho $ , where $\bar \rho $ is a small positive number and it can be chosen such that $4C_0\bar \rho j^2=1$ , where $C_0$ is a uniform constant, specified below. By choosing $\bar \rho $ , we require the derivatives of $\rho $ satisfying

(3.10) $$ \begin{align}|\rho^{'}|+|\rho^{"}|\leq 4C_0\bar\rho j^2= 1.\end{align} $$

For $x\in B_2$ , we also denote $\rho (x)=\rho (|x|)$ . The choice of the function $\rho $ depends crucially on the cutoff function $\psi _j$ , and this is the first key point in our construction.

Given such a function $\rho $ , we construct, for $x\in B=B_1$ ,

(3.11) $$ \begin{align} J_{k,\, j,\,\rho(x)}(x)=\int_{B_{\rho(x)}(x)}\phi_\rho(y-x)J_{k,\, j}(y)dy=\int_B \phi(z) J_{k,\, j}(x+\rho(x)z)dz. \end{align} $$

By the choice of $\rho (x)$ , which is zero when $1\leq |x|\leq 2$ or $|x|\leq 1-j^{-1}$ , $J_{k, j, \rho (x)}$ can be extended to M and satisfies (3.7). Similarly, we denote

(3.12) $$ \begin{align} &J_{0,\, \rho(x)}(x)=\int_{B_{\rho(x)}(x)}\phi_\rho(y-x)J_0(y)dy=\int_B \phi(z)J_0(x+\rho(x)z)dz. \nonumber \\ &J_{k,\,\rho(x)}(x)=\int_{B_{\rho(x)}(x)}\phi_\rho(y-x)J_k(y)dy=\int_B \phi(z)J_k(x+\rho(x)z)dz. \end{align} $$

Note that $J_{0,\,\rho (x)}(x)$ , $J_{k, \, \rho (x)}(x)$ and $J_{k,\, j,\,\rho (x)}(x)$ are neither almost complex structures nor compatible with the metric g in general. It is straightforward to see that $J_{0,\,\rho (x)}(x)$ is close to $J_0(x)$ given $\bar \rho $ is sufficiently small ( $J_0$ is smooth and $J_{0, \, \rho }$ is a standard approximation). We will also show that $J_{k, \,\rho (x)}(x)$ is close to an almost complex structure and it is almost compatible with the metric g, using the Poincare inequality and small energy assumption. One difficulty is to prove that $J_{k,\, j, \,\rho }$ is also close to an almost complex structure, with a suitable choice of $\rho $ and $\bar \rho $ and $k=k(\bar \rho , j)$ . The dependence of k on $\bar \rho $ and j is inevitable. In particular, we need to choose $\rho $ depending on $\psi _j$ .

Once we construct $J_{k,\, j, \,\rho }$ , we use the technique in [Reference He5] to construct a unique almost complex structure $\tilde J_k$ using $J_{k,\, j,\, \rho }$ , such that it is compatible with g. We assert that

(3.13) $$ \begin{align} \int_M |\nabla J_k-\nabla \tilde J_k|^4dv\leq C \epsilon_0 \end{align} $$

which implies that $\tilde J_k\in \sigma $ using a theorem of White [Reference White15, Theorem 2 and Section 6].

Here comes another essential point of the proof. By the construction, $\tilde J_k$ equals $J_k$ outside $B_1$ and agrees with $J_0$ in $B_{1-j^{-1}}$ . If $\tilde J_k$ is an energy-minimizing sequence in $\sigma $ , this implies that $\lambda \equiv 0$ in $B_{1}$ and completes the proof. Unfortunately, we are not able to prove

(3.14) $$ \begin{align} \int_{B_1\setminus B_{1-j^{-1}}}|\Delta \tilde J_{k}|^2dv\rightarrow 0 \end{align} $$

for $k\geq k_j$ and $j\rightarrow \infty $ . Instead, we prove the following inequality approximately.

(3.15) $$ \begin{align} \int_{B_1\setminus B_{1-j^{-1}}}|\Delta \tilde J_{k}|^2dv\lessapprox C \int_{B_{1}\setminus B_{1-j^{-1}}} (|\nabla^2 J_k|^2+|\nabla J_k|^4)dv. \end{align} $$

Given (3.15) and the fact that $\{J_k\}$ is an energy-minimizing sequence, we can obtain that

(3.16) $$ \begin{align} \lambda(B_1)\leq C\nu (\partial B_1). \end{align} $$

Since the defect measure $\nu (\partial B_1)$ can be strictly positive on $\partial B_1$ , this does not directly lead to the conclusion $\lambda \equiv 0$ . However, the construction above actually works on any ball $B_r\subset B_2$ , $r\in [1/4, 7/4]$ (replacing $B_1$ by $B_r$ ), and the arguments can be directly carried over. Hence, we will prove for $r\in [1/4, 7/4]$ ,

(3.17) $$ \begin{align} \lambda(B_r)\leq C\nu (\partial B_r). \end{align} $$

In particular, we have for $r\in [3/2, 7/4]$ ,

(3.18) $$ \begin{align} \lambda(B_{3/2})\leq C\nu(\partial B_r). \end{align} $$

Since $\nu $ is a totally bounded positive Radon measure, then $\nu (\partial B_r)=0$ for infinitely many r (actually $\nu (\partial B_r)>0$ for at most countably many r). Hence, this proves that $\lambda \equiv 0$ in $B_{3/2}$ . It is then a standard practice to prove that $\nu \equiv 0$ in $B_1$ .

We state two versions of the Poincare inequality which are needed in the proof. For $f\in W^{1, 2}(B_R), B_R\subset \mathbb {R}^n$ , denote $\underline {f}$ to be its average in the ball,

$$\begin{align*}\underline{f}=\frac{1}{\text{Vol}(B_R)}\int_{B_R} f(y)dy. \end{align*}$$

Then we have

(3.19) $$ \begin{align} R^{-n}\int_{B_R}|f-\underline{f}|^2dy\leq C R^{2-n}\int_{B_R}|Df|^2dy, \end{align} $$

where C is a uniform dimensional constant. Suppose $\phi $ is a cutoff function supported in $B_R$ such that $\int _{B_R}\phi (y)dy=1$ . We denote

$$\begin{align*}f_{*}=\int_{B_R} \phi (y) f(y)dy, \end{align*}$$

and then we have

(3.20) $$ \begin{align} R^{-n}\int_{B_R}|f-f_{*}|^2dy\leq C R^{2-n}\int_{B_R}|Df|^2dy. \end{align} $$

We should mention that f can be taken as vector-valued and matrix-valued functions as a direct generalization. Now we carry out the details to prove Lemma 3.9.

Proof. Step one: the construction of an ‘almost’ almost complex structure $J_{k, j, \rho (x)}$ .

Note that J is g-compatible if $J+J^*=0$ . With the localization over $B_2$ , J is g-compatible if $(J+g J^t g^{-1})(x)=0$ holds as a matrix-valued equation for all $x\in B_2$ , where $J^t$ is the transpose of J. If J is compatible with g, then $J_\rho $ is almost g-compatible if $\bar \rho $ is sufficiently small (we assume that $|J|$ is bounded, of course). We estimate

(3.21) $$ \begin{align} |(J_\rho+gJ_\rho^t g^{-1})(x)|=&\left|\int_{B_\rho(x)}\phi_\rho(y-x) (J(y)+g(x)J^t(y)g^{-1}(x))dy\right| \nonumber \\ \leq &\rho^{-4}\int_{B_\rho}\left|J(y)+g(x)J^t(y)g^{-1}(x)\right|dy \nonumber \\ \leq &\rho^{-4}\int_{B_\rho} \left|-g(y)J^t(y)g^{-1}(y)+g(x)J^t(y)g^{-1}(x)\right|dy \nonumber \\ \leq& C\delta_0 \bar \rho, \end{align} $$

where we use the facts $|g(x)-g(y)|\leq C\delta _0\bar \rho \,\, \mbox {for}\,\, |y-x|\leq \rho $ and

$$ \begin{align*} &g(x)J^t(y)g^{-1}(x)-g(y)J^t(y)g^{-1}(y)=(g(x)-g(y)) J^t(y)g^{-1}(x)+g(x)J^t(y)(g^{-1}(x)-g^{-1}(y)) \end{align*} $$

It is clear that (3.21) holds for all the cases $J=J_0,\, J_k\, \mbox {and}\, J_{k,\, j}$ .

Next we show that $J_{k,\, j,\,\rho }$ is ‘almost’ an almost complex structure in the sense that $|J_{k,\, j,\,\rho }J_{k,\,j,\,\rho }+\text {id}|$ is very small pointwise. Now we specify $\rho $ further and discuss the properties of $J_{k,\, j,\,\rho }$ . For ${|x|\in [0, 1-j^{-1}]\cup [1, 2]}$ , since $\rho (x)=0$ , we have

(3.22) $$ \begin{align} J_{k,\,j, \,\rho(x)}(x)=J_{k,\, j}(x)=\begin{cases} &\!\!\!\! J_k(x), |x|\in [1, 2],\\ &\!\!\!\! J_0(x), |x|\in [0, 1-j^{-1}].\end{cases}\end{align} $$

Fix $\delta _1>0$ sufficiently small (which can be taken as $j^{-1}$ ). We write $(1-j^{-1}, 1)$ as three subintervals

$$\begin{align*}(1-j^{-1}, 1-j^{-1}+\delta_1 j^{-1}]\cup (1-j^{-1}+\delta_1 j^{-1}, 1-\delta_1 j^{-1})\cup [1-\delta_1 j^{-1}, 1).\end{align*}$$

The discussions in each subinterval are different. We choose $\rho $ such that

(3.23) $$ \begin{align} \begin{cases} &\!\!\!\! \rho(1-(2j)^{-1})=\bar \rho\\ &\!\!\!\! \rho(1-j^{-1}+\delta_1 j^{-1})=\rho(1-\delta_1 j^{-1})=\delta_1 \bar\rho\\ &\!\!\!\! \rho(s)<\delta_1\bar \rho, s\in (1-j^{-1}, 1-j^{-1}+\delta_1 j^{-1})\cup (1-\delta_1 j^{-1}, 1)\\ &\!\!\!\! \rho(s)\geq \delta_1\bar \rho, s\in (1-j^{-1}+\delta_1 j^{-1}, 1-\delta_1 j^{-1}). \end{cases} \end{align} $$

Let $h(s)$ be a smooth nonnegative function supported over $[0, 2]$ , satisfying the following.

(3.24) $$ \begin{align} &h(1)=1=\max h, \text{and}\; h(s)=h(2-s), \nonumber \\ &h(j^{-1})=j^{-1}, \; h'(s)\geq 0, s\in [0, 1], \nonumber \\ &|h^{'}|+|h^{"}|\leq C_0. \end{align} $$

Denote $\rho (s)= \bar \rho \; h(2+2j(s-1))$ . Then $\rho (s)$ is supported in $[1-j^{-1}, 1]$ , satisfying (3.23). We now choose $\bar \rho $ such that

$$\begin{align*}|\rho^{'}|+|\rho^{"}|\leq 4\bar \rho j^2 (|h^{'}|+|h^{"}|)\leq 4C_0 \bar\rho j^2=1. \end{align*}$$

For $|x|\in (1-j^{-1}, 1-j^{-1}+\delta _1 j^{-1}]$ , we have

$$\begin{align*}|x+\rho(x)z|\leq 1-j^{-1}+\delta_1 j^{-1}+\delta_1\bar \rho.\end{align*}$$

Using $\psi _j(1-j^{-1})=0$ and $|\psi ^{'}_j|\leq 3j$ , we get

$$\begin{align*}\psi_j(|x+\rho(x)z|)\leq 3j(\delta_1 j^{-1}+\delta_1\bar \rho)\leq 4\delta_1.\end{align*}$$

We compute

$$ \begin{align*} |J_{k,\, j,\,\rho(x)}(x)-J_{0,\, \rho(x)}(x)|=\left|\int_B\phi(z)\psi_j(|x+\rho(x)z)|(J_0-J_k)(x+\rho(x)z)dz\right|\leq 100 \delta_1. \end{align*} $$

Similarly, for $|x|\in [1-\delta _1 j^{-1}, 1)$ , we have

(3.25) $$ \begin{align} |J_{k,\, j,\,\rho(x)}(x)-J_{k,\, \rho(x)}(x)|\leq 100\delta_1. \end{align} $$

We can estimate

(3.26) $$ \begin{align} |J_{0,\,\rho(x)}(x)-J_0(x)|\leq \int_B \phi(z)|J_0(x+\rho(x)z)-J_0(x)|dz\leq C_1 \bar\rho, \end{align} $$

where $C_1=C_1(\max |\nabla J_0|)$ is a uniform constant. We derive

(3.27) $$ \begin{align} |J_{k,\, j,\,\rho(x)}-J_0(x)|\leq 100\delta_1+C_1 \bar \rho, |x|\in (1-j^{-1}, 1-j^{-1}+\delta_1 j^{-1}]. \end{align} $$

This implies that $J_{k,\, j,\,\rho (x)}$ is close to an almost complex structure for $|x|\in (1-j^{-1}, 1-j^{-1}+\delta _1 j^{-1})$ , provided that $100\delta _1+C_1 \bar \rho $ is sufficiently small. Since $|\nabla J_k|$ might not be uniformly bounded, we do not have an effective pointwise estimate on $|J_{k,\,\rho (x)}(x)-J_k(x)|$ as above. Instead, we apply the Poincare inequality (3.20) in the ball $B_\rho (x)$ to $J(y)$ with $ J_{*}=J_\rho (x)=\int _{B_\rho (x)} \phi _\rho (y-x) J(y)dy, $

(3.28) $$ \begin{align} \rho^{-4}\int_{B_{\rho}(x)} |J(y)-J_\rho(x)|^2 dv\leq C\rho^{-2}\int_{B_\rho(x)}|DJ(y)|^2dy, \end{align} $$

Replacing J by $J_k$ , we can get that

(3.29) $$ \begin{align} \rho^{-4}\int_{B_\rho(x)}|J_k(y)-J_{k,\,\rho(x)}(x)|^2dy\leq C\rho^{-2}\int_{B_\rho(x)}|DJ_k(y)|^2dy. \end{align} $$

By the Hölder inequality, we see that

$$\begin{align*}\rho^{-2}\int_{B_\rho(x)}|DJ_k(y)|^2dy\leq C\left(\int_{B_\rho(x)}|DJ_k|^4dy\right)^{\frac{1}{2}}. \end{align*}$$

Using the fact that $\nabla J=DJ+\partial g*J$ and (3.5), it follows that

(3.30) $$ \begin{align} \rho^{-4}\int_{B_\rho(x)}|J_k(y)-J_{k,\,\rho(x)}(x)|^2dy\leq C\left(\int_{B_\rho(x)}|DJ_k|^4dy\right)^{\frac{1}{2}}\leq C(\sqrt{\epsilon_0}+\sqrt{\delta_0}). \end{align} $$

Hence, (3.30) implies that there are many ys in $B_{\rho }(x)$ such that

$$\begin{align*}|J_k(y)-J_{k,\, \rho(x)}(x)|\leq C(\sqrt[4]{\epsilon_0}+\sqrt[4]{\delta_0}). \end{align*}$$

In particular, this implies that

$$\begin{align*}|J_{k,\, \rho(x)}(x)J_{k,\, \rho(x)}(x)+\text{id}|\leq C(\sqrt[4]{\epsilon_0}+\sqrt[4]{\delta_0}). \end{align*}$$

Using (3.25) and the above, we get that

(3.31) $$ \begin{align} |J_{k,\, j,\,\rho(x)} J_{k,\, j,\,\rho(x)}+\text{id}|\leq C(\delta_1+\sqrt[4]{\epsilon_0}+\sqrt[4]{\delta_0}), |x|\in [1-\delta_1 j^{-1}, 1). \end{align} $$

Next we consider $|x|\in (1-j^{-1}+\delta _1j^{-1}, 1-\delta _1 j^{-1})$ , where $\rho (x)\geq \delta _1\bar \rho $ . We compute

(3.32) $$ \begin{align} &|J_{k,\, j,\,\rho(x)}(x)-J_{0,\, \rho(x)}(x)|\leq \int_B \phi(z) |(J_0-J_k)(x+\rho(x)z)|dz. \end{align} $$

We have

$$\begin{align*}\int_B \phi(z) |(J_0-J_k)(x+\rho(x)z)|dz\leq &{\rho(x)}^{-4} \int_{B_{\rho(x)}(x)}|J_0(y)-J_k(y)|dy \nonumber \\ \leq&C\rho^{-2}\|J_0-J_k\|_{L^2(B_{3/2})}. \nonumber\end{align*}$$

Since $J_k$ converges to $J_0$ strongly in $W^{1, 2}$ and $\rho \geq \delta _1\bar \rho $ for $|x|\in (1-j^{-1}+\delta _1j^{-1}, 1-\delta _1 j^{-1})$ , we can choose $k_0=k_0(\delta _1, \bar \rho )$ sufficiently large such that

$$\begin{align*}C\rho^{-2} \|J_0-J_k\|_{L^2}\leq \bar\rho. \end{align*}$$

Hence, we get, for $k\geq k_0$ ,

(3.33) $$ \begin{align} {\rho}^{-4} \int_{B_{\rho(x)}(x)}|J_0(y)-J_k(y)|dy\leq \bar\rho. \end{align} $$

Using (3.33) and (3.32), we have that, for $k\geq k_0$ ,

$$\begin{align*}|J_{k,\, j,\,\rho(x)}(x)-J_{0,\, \rho(x)}(x)|\leq \bar \rho. \end{align*}$$

This together with (3.26) implies that, for $k\geq k_0$ ,

$$\begin{align*}|J_{k,\, j,\,\rho(x)}(x)-J_0(x)|\leq C\bar\rho, \end{align*}$$

and in particular, we have

$$\begin{align*}|J_{k,\, j,\,\rho(x)}(x)J_{k,\, j\,, \rho(x)}(x)+\text{id}|\leq C \bar \rho. \end{align*}$$

Step two: construction of almost complex structure $\tilde J_k$ by projecting $J_{k, \;j, \;\rho (x)}$ w.r.t g.

Next we construct a sequence $\tilde J_k(x)$ using $J_{k,\, j, \,\rho (x)}(x)$ by the technique we have used in [Reference He5, see (4.15), (4.16)]. We briefly recall the construction. In the following, we consider $k\geq k_j$ for each j. For $|x|\in [0, 1-j^{-1}]\cup [1, 2]$ , $\rho =0$ , and we have

(3.34) $$ \begin{align}\tilde J_k=J_{k,\, j, \, \rho}=J_{k,\, j}= \begin{cases} J_0,\, |x|\leq 1-j^{-1}\\ J_k,\, |x|\geq 1 \end{cases}\end{align} $$

Now consider $|x|\in (1-j^{-1}, 1)$ . Let $S_{g}(x)$ and $A_{g}(x)$ be the g-symmetric and g-skew symmetric part of $J_{k,\, j, \,\rho (x)}(x)$ , respectively. We have by (3.21)

$$\begin{align*}|S_{g}(x)|=\frac{1}{2} |(J_{k,\,j, \,\rho(x)}(x)+g(x)J^t_{k,\,j, \,\rho(x)}(x)g^{-1}(x))|\leq C\delta_0\bar \rho. \end{align*}$$

It follows that

$$\begin{align*}|A_{g}^2+\text{id}|<C\sqrt[4]{\epsilon_0}+C\delta_0 {\bar\rho}.\end{align*}$$

Note that $-A_{g}^2$ is g-symmetric and that it is close to the identity matrix pointwise; in particular, it is positive definite. Denote $Q_{g}$ to be the g-symmetric matrix such that $Q_{g}^2=-A_{g}^2$ . Note that $Q_{g}$ is uniquely determined and it commutes with $A_g$ . Denote $\tilde J_{k}(x)=Q_{g}^{-1}(x)A_{g}(x)$ for $x\in B_2$ . Then $\tilde J_{k}(x)$ is a g-compatible almost complex structure in $B_2$ . We extend $\tilde J_k$ to M by simply putting $\tilde J_k=J_k$ on $M\setminus B_{1}(p)$ . Now we establish (3.13). We only need to consider over $B_1(p)$ . Note that the $L^\infty $ norm of $J_{k,\, j}, Q_g, A_g, Q_g^{-1}$ and $1/|Q_{g}^{-1}|$ are all uniformly bounded by a dimensional constant. Since $Q_g^2=-A_g^2$ and $|A_g^2+\text {id}|\ll 1$ , we have the following expansion of the matrix.

(3.35) $$ \begin{align} Q_g=\sqrt{\text{id}-(\text{id}+A_g^2)}=\sum_{l=0}^\infty \binom{1/2}{l}(\text{id}+A_g^2)^l. \end{align} $$

We can compute directly that $|\nabla Q_g|\leq C|A_g||\nabla A_g|\leq C|\nabla A_g|$ . Hence, we obtain $|\nabla \tilde J_k|\leq C|\nabla A_g|.$ We also need (using $\nabla g=0$ )

$$\begin{align*}|\nabla S_g|=|\nabla J_{k, j, \rho}+g \nabla J^{t}_{k, j, \rho} g^{-1}|\leq C |\nabla J_{k, j, \rho}|. \end{align*}$$

We compute

$$\begin{align*}|\nabla J_{k, j, \rho}|\leq C\int_B \phi(z) |\nabla J_{k, j}(x+\rho(x)z)| (1+|\rho^{'}|)dz. \end{align*}$$

Hence, we have

$$\begin{align*}|\nabla J_{k, j, \rho}|^4\leq C \int_B |\nabla J_{k, j}(x+\rho(x)z)|^4dz. \end{align*}$$

Moreover, we have $|\nabla A_g|\leq |\nabla J_{k, j, \rho }|+|\nabla S_g|\leq C|\nabla J_{k, j, \rho }|$ . It follows that

$$\begin{align*}\int_B |\nabla \tilde J_k|^4dv\leq C\int_B \left(\int_B |\nabla J_{k, j}(x+\rho(x)z)|^4dz\right)dv_x\leq C \int_{B_{3/2}} |\nabla J_{k, j}|^4dv. \end{align*}$$

Since $\nabla J_{k, j}=\nabla J_k+(1-\psi _j)(\nabla J_0-\nabla J_k)-(J_0-J_k) \nabla \psi _j$ and $|\nabla \psi _j|\leq 3j$ , it follows that

(3.36) $$ \begin{align} \int_{B_{3/2}} |\nabla J_{k, j}|^4dv\leq C \int_{B_2} (|\nabla J_k|^4+|\nabla J_0|^4) dv+C j^4\int_B |J_0-J_k|^4dv. \end{align} $$

Using the Sobolev inequality, we know that

$$\begin{align*}\|J_0-J_k\|_{L^4}\leq C\|J_0-J_k\|_{W^{1, 2}}. \end{align*}$$

By choosing $k_j=k(j)$ sufficiently large such that for $k\geq k_j$ , we can assume that

(3.37) $$ \begin{align} Cj^4\int_{B_1}(|J_0-J_k|^2+|J_0-J_k|^4+|\nabla J_0-\nabla J_k|^2)dv \leq j^{-1}\leq \epsilon_0. \end{align} $$

This establishes (3.13) and hence $\tilde J_k$ and $J_k$ are in the same homotopy class, for $k\geq k_j$ .

Step three: the comparison of $J_k$ and $\tilde J_k$ implies $\lambda (B_1)\leq C\nu (\partial B_1)$ .

Fix $\epsilon>0$ . Since $J_k$ is an energy-minimizing sequence, for k sufficiently large we have,

$$\begin{align*}\int_M |\Delta J_k|^2dv\leq \int_M |\Delta \tilde J_k|^2 dv+\epsilon. \end{align*}$$

By the construction of $\tilde J_k$ , we get

$$ \begin{align*} \int_{B_1}|\Delta J_k|^2 dv\leq \int_{B_{1}\setminus B_{1-j^{-1}}}|\Delta \tilde J_k|^2dv+\int_{B_{1-j^{-1}}} |\Delta J_0|^2 dv+\epsilon. \end{align*} $$

By taking $j\rightarrow \infty $ (hence $k\geq k_j\rightarrow \infty $ ), we get (since $B_1$ is open)

$$ \begin{align*} \lambda(B_1)+\int_{B_{1}}|\Delta J_0|^2 dv\leq \liminf \int_{B_{1}\setminus B_{1-j^{-1}}}|\Delta \tilde J_k|^2dv+\int_{B_{1-j^{-1}}} |\Delta J_0|^2 dv+\epsilon. \end{align*} $$

Hence, we get

$$ \begin{align*} \lambda(B_1)\leq \liminf_{j\rightarrow \infty} \int_{B_{1}\setminus B_{1-j^{-1}}}|\Delta \tilde J_k|^2dv+\epsilon. \end{align*} $$

Since $\epsilon>0$ is arbitrary, we have established the estimate

(3.38) $$ \begin{align} \lambda(B_1)\leq \liminf_{j\rightarrow \infty} \int_{B_{1}\setminus B_{1-j^{-1}}}|\Delta \tilde J_k|^2dv. \end{align} $$

Now we need estimates as in (3.15) to control the right-hand side of (3.38). Recall that we have the unique decomposition $J_{k,\, j, \,\rho }=A_g+S_g$ and $\tilde J_k=Q_g^{-1}A_g$ , where $Q_g$ is the unique square root of $-A_g^2$ . Using (3.35), we have

$$\begin{align*}|\nabla Q_g|\leq C|\nabla A_g|, |\Delta Q_g|\leq C(|\Delta A_g|+|\nabla A_g|^2). \end{align*}$$

It follows that

$$\begin{align*}|\Delta (Q^{-1}_g A_g)|\leq C(|\Delta A_g|+|\nabla A_g|^2)\leq C(|\Delta J_{k, \, j, \,\rho}|+|\nabla J_{k,\, j, \,\rho}|^2). \end{align*}$$

Hence, we obtain

(3.39) $$ \begin{align} \int_{B_{1}\setminus B_{1-j^{-1}}}|\Delta \tilde J_k|^2dv\leq C\int_{B_{1}\setminus B_{1-j^{-1}}}\left( |\Delta J_{k, \, j, \,\rho}|^2+|\nabla J_{k,\, j, \,\rho}|^4\right)dv. \end{align} $$

We compute

$$ \begin{align*} |\nabla J_{k,\,j}|=&|\nabla J_k+(1-\psi_j)(\nabla J_0-\nabla J_k)-\nabla \psi_j (J_0-J_k)| \nonumber \\ =&|\psi_j \nabla J_k+(1-\psi_j)\nabla J_0-\nabla \psi_j (J_0-J_k)| \nonumber \\ \leq& |\nabla J_k|+|\nabla J_0|+Cj |J_0-J_k|. \end{align*} $$

Similarly, we compute

$$ \begin{align*} |\Delta J_{k,\,j}|=&|\Delta J_k+\Delta [(1-\psi_j)( J_0-J_k)]| \nonumber \\ \leq& |\nabla^2 J_k|+|\nabla^2 J_0|+Cj^2 (|J_0-J_k|+|\nabla J_0-\nabla J_k|). \end{align*} $$

Write $y=x+\rho (x)z$ . Then we have

$$\begin{align*}\left|\frac{\partial y_i}{\partial x_j}\right|\leq C(1+|\rho^{'}|), \left|\frac{\partial^2 y_i}{\partial x_j \partial x_k}\right|\leq C(1+|\rho^{'}|+|\rho^{"}|). \end{align*}$$

Using $|\rho ^{'}|+|\rho ^{"}|\leq 1$ , $|\psi ^{'}_j|+|\psi ^{"}_j|\leq 20j^2$ , we can then get

$$ \begin{align*} |\nabla J_{k,\, j, \,\rho}|=&\left|\int_B \phi(z) \nabla_x J_{k,\, j}(x+\rho(x)z)dz\right|\nonumber \\ \leq &C\int_B \phi(z)|\nabla_z J_{k,\, j}| (1+|\rho^{'}|)dz\nonumber \\ \leq& C\int_{B_1} \phi(z)\left(|\nabla J_k|+|\nabla J_0|+j |J_0-J_k|\right)dz, \end{align*} $$

where the function is evaluated at $y=x+\rho (x)z$ . For any open set $U\subset B_2$ , denote $U_{\bar \rho }=\{x: \text {dist}(x, U)<\bar \rho \}$ . We have

(3.40) $$ \begin{align} \int_{U}|\nabla J_{k,\, j, \,\rho}|^4dv_x\leq& C\int_{U}\left(\int_{B_1} \phi(z)\left(|\nabla J_k|(y)+|\nabla J_0|(y)+j |J_0-J_k|(y)\right)dz\right)^4dv_x \nonumber \\ \leq &C\int_{U_{\bar\rho}} (|\nabla J_k|^4+|\nabla J_0|^4+j^4 |J_0-J_k|^4)dv, \end{align} $$

where we have used a standard technique to estimate the $L^p$ norm of mollifier approximation (see [Reference Gilbarg and Trudinger4, Lemma 7.2, (7.15)]). Similarly, we compute

$$ \begin{align*} |\Delta J_{k,\, j, \,\rho}|=&\left|\int_B \phi(z) \Delta_x \left(J_{k,\, j}(x+\rho(x)z)\right)dz\right| \nonumber \\ \leq &C\int_B \phi(z)\left(|\nabla^2 J_k|+|\nabla^2 J_0|+j^2(|\nabla J_0-\nabla J_k|+|J_0-J_k|)\right) dz. \end{align*} $$

It follows that

(3.41) $$ \begin{align} \int_{U}|\Delta J_{k,\, j, \,\rho}|^2dv_x\leq C \int_{U_{\bar \rho}} \left(|\nabla^2 J_k|^2+|\nabla^2 J_0|^2+j^4|J_0-J_k|^2+j^4|\nabla J_0-\nabla J_k|^2\right)dv_x. \end{align} $$

Take $U=B_1\setminus B_{1-j^{-1}}$ in the above. We obtain,

(3.42) $$ \begin{align} \int_{U} (|\nabla J_{k,\, j, \,\rho}|^4+|\nabla^2 J_{k,\, j, \,\rho}|^2)dv\leq C\int_{U_{\bar \rho}} (|\nabla^2 J_k|^2+|\nabla J_k|^4)dv+C(R_1+R_2), \end{align} $$

where the remainder terms read,

$$ \begin{align*} R_1=&\int_{U_{\bar \rho}} (|\nabla J_0|^4+|\Delta J_0|^2 )dv, \nonumber \\ R_2=&j^4\int_{U_{\bar \rho}} (|J_0-J_k|^2+|J_0-J_k|^4+|\nabla J_0-\nabla J_k|^2)dv. \end{align*} $$

Recall we assume that $k\geq k_j$ , such that (3.37) holds. Hence, $R_2\leq j^{-1}$ . Certainly when $j\rightarrow \infty $ , the Lebesgue measure $U_{\bar \rho }\rightarrow 0$ and hence $R_1\rightarrow 0$ . By (3.42), we obtain

(3.43) $$ \begin{align} \liminf_{j\rightarrow \infty}\int_{B_1\setminus B_{1-j^{-1}}}(|\nabla J_{k,\, j, \,\rho}|^4+|\Delta J_{k,\, j, \,\rho}|^2)dv\leq C\liminf_{j\rightarrow\infty}\int_{U_{\bar \rho}} (|\nabla^2 J_k|^2+|\nabla J_k|^4)dv. \end{align} $$

Hence, we obtain, by (3.39) and (3.43),

(3.44) $$ \begin{align} \liminf_{j\rightarrow \infty}\int_{B_{1}\setminus B_{1-j^{-1}}}|\Delta \tilde J_k|^2dv\leq C\liminf_{j\rightarrow \infty}\int_{U_{\bar\rho}} (|\nabla^2J_k|^2+|\nabla J_k|^4)dv. \end{align} $$

Note that $U=B_{1}\setminus B_{1-j^{-1}}$ , $\bar \rho =j^{-2}/10$ . Fix $\epsilon>0$ . Denote

$$\begin{align*}B_{1,\,\epsilon}=\{x: \text{dist}(x, \partial B_1)<\epsilon\}. \end{align*}$$

For any j sufficiently large, $U_{\bar \rho }\subset \overline {B_{1,\,\epsilon }}$ . Hence, we have

$$\begin{align*}\lim_{k\rightarrow\infty}\int_{\overline{B_{1,\,\epsilon}}} (|\nabla^2J_k|^2+|\nabla J_k|^4)dv=\lim_{k\rightarrow\infty}\mu_k(\overline{B_{1,\,\epsilon}})\leq \mu(\overline{B_{1,\,\epsilon}}), \end{align*}$$

where we use the fact that $\mu _k$ converges to $\mu $ weakly and $\overline {B_{1,\,\epsilon }}$ is a closed set. Together with (3.44), we get that

(3.45) $$ \begin{align} \lim_{j\rightarrow \infty}\int_{B_{1}\setminus B_{1-j^{-1}}}|\Delta \tilde J_k|^2dv\leq C\mu(\overline{B_{1,\,\epsilon}}). \end{align} $$

Note that $\cap _{\epsilon>0} \overline {B_{1,\,\epsilon }}=\partial B_1$ . We obtain $\lim _{\epsilon \rightarrow 0}\mu (\overline {B_{1,\,\epsilon }})=\nu (\partial B_1).$ With (3.45), we obtain

(3.46) $$ \begin{align} \lim_{j\rightarrow \infty}\int_{B_{1}\setminus B_{1-j^{-1}}}|\Delta \tilde J_k|^2dv\leq C\nu(\partial B_1). \end{align} $$

Hence, we have obtained the desired estimate

(3.47) $$ \begin{align} \lambda(B_1)\leq C \nu(\partial B_1). \end{align} $$

If we replace $B_1$ by $B_r$ , for $1\leq r\leq \frac {3}{2}$ , replace $B_{1}\setminus B_{1-j^{-1}}$ by $B_{r}\setminus B_{r-j^{-1}}$ and apply the same arguments as in the proof (3.47), we obtain

(3.48) $$ \begin{align} \lambda(B_r)\leq C\nu(\partial B_r). \end{align} $$

Since $\nu $ is totally bounded, this implies that $\lambda (B_r)\equiv 0$ for $r\leq 3/2$ . This completes the proof of Lemma 3.9.

With Lemma 3.9, we prove Lemma 3.7, which asserts that the homotopy class of $J_0$ is either $\sigma $ or $p(\sigma )$ , for a minimizing sequence $J_k$ in $\sigma $ .

Proof of Lemma 3.7.

Lemma 3.9 implies that $J_k$ converges to $J_0$ strongly in $W^{2, 2}(M\backslash U_{r/2})$ , where U is the collection of finitely many disjoint geodesic balls $B(p_i)$ , $i=1, \cdots N$ with radius $r/2$ , for any sufficiently small positive number r. Moreover, for any point $p\in M\backslash U_r$ , we have that

$$\begin{align*}\int_{B_{r/2}(p)} |\nabla J_k|^4+|\nabla^2 J_k|^2 \leq \epsilon_0. \end{align*}$$

Hence, we can construct $\tilde J_k$ such that

$$ \begin{align*} &\tilde J_k(p)= J_0(p), p\in M\backslash U_{2r}, \nonumber \\ &\tilde J_k(p)=J_k(p), p\in U_{r/2} \end{align*} $$

and

$$\begin{align*}\int_M |\nabla J_k-\nabla \tilde J_k|^4\leq C\epsilon_0. \end{align*}$$

This implies that $\tilde J_k$ is in $\sigma $ . In short, we construct $\tilde J_k$ in the same homotopy class which coincides $J_0$ over $M\backslash U_{2r}$ , where the convergence is strongly in $W^{2, 2}$ .

Given a homotopy class $\sigma $ of almost complex structures on M, we recall the construction of $p(\sigma )$ (see Donaldson [Reference Donaldson3, Section 6]). A compatible almost complex structure J on an oriented Riemannian four-manifold M can be considered as a section of the associated $SO(4)/U(2)$ -bundle over M (the sphere bundle of $\Lambda ^2_+$ , known as the ‘twistor space’). Suppose there are two almost complex structures $J_1$ and $J_2$ , which agree each other outside a small ball in M. Over this ball B, $J_1$ and $J_2$ compare by a map from $S^4$ to the fibre of the twistor bundle, hence defining an element in $[S^4, S^2]=\mathbb {Z}/2$ . If $\sigma $ is a homotopy class, $p(\sigma )$ agrees with $\sigma $ outside a small ball, and over the ball the two compare by the nonzero element of $[S^4, S^2]=\mathbb {Z}/2$ . Hence, if $J_1$ and $J_2$ agree outside a small ball, then either they are in the same homotopy class or their homotopy classes are related by the map p.

Applying this to $\tilde J_k$ and $J_0$ above, the homotopy classes differ by the composition $p^k$ , for some $k\leq N$ . Since $p\circ p=id$ , it follows that $J_0$ is either in $\sigma $ or $p(\sigma )$ .

Theorem 1.3 follows as an immediate consequence of Lemma 3.7.

4 An intuitive conjectural picture

A $K3$ surface is a compact simply connected complex surface S with trivial canonical bundle. Let $\sigma $ denote the homotopy class of the standard complex structures on S with $c_1=0$ . S. Donaldson [Reference Donaldson3, Corollary 6.5] proved, using his polynomial invariants, the following.

Theorem 1.3 does not specify precisely whether $\sigma $ or $p(\sigma )$ contains an energy-minimizing biharmonic almost complex structure. There could be a couple options, as follows.

1. 1. Both $\sigma $ and $p(\sigma )$ contain an energy-minimizing biharmonic almost complex structure, which might or might not have the same energy.

2. 2. Only one homotopy contains an energy-minimizing biharmonic almost complex structure, while the other does not.

We believe the following and Theorem 4.1 serves as an example.

In the process of energy-minimizing, the formation of a bubble gives a nonconstant extrinsic biharmonic map from $\mathbb {R}^4$ to $S^2$ by a blowup argument. But it does not seem to be straightforward to specify its relative homotopy class, even though it is intuitive that it should correspond to the nonzero element in $\pi _4(S^2)$ topologically. We believe the following.