The existence of J-holomorphic curves in almost Hermitian manifolds

Abstract

In this paper, we investigate the existence of J-holomorphic curves on almost Hermitian manifolds. Let (M, g, J, F) be an almost Hermitian manifold and $f:\Sigma \rightarrow M$ be an injective immersion. We prove that if the $L_p$ functional has a critical point or a stable point in the same almost Hermitian class, then the immersion is J-holomorphic.

1 Introduction

Let (M, J) be a closed almost complex 2n-manifold and $\Sigma$ be a closed real surface. We call a smooth immersion $f:\Sigma \rightarrow M$ J-holomorphic if $J_{f(p)}$ maps $f_{*p}(T_p\Sigma )$ onto itself for any point $p\in \Sigma$. Under what condition an immersion is J-holomorphic is an interesting question in differential geometry. Recently, Arezzo and Sun [1] gave a variational characterization of J-holomorphic curves in almost Kähler manifold $(M,g,J,\omega )$. More precisely, they consider the change of the area functional according to the change of the symplectic form on M in the fixed cohomology class (with fixed immersion f and fixed almost complex structure J on M). Let

$$\begin{aligned} \mathcal {\tilde{H}}\triangleq \left\{ \rho \in C^\infty (M,\mathbb {R}):\omega _\rho \triangleq \omega +dd^c_J\rho \,\,\,\mathrm{tames}\,\,\,J\right\} , \end{aligned}$$

which is clearly a nonempty open subset of $C^\infty (M,\mathbb {R})$. Here, $d^c_J$ is the twisted exterior differential defined by $d^c_J=(-1)^pJdJ$ acting on p-forms, in particular $d^c_J\rho (X)=-d\rho (JX)$. To each $\rho \in \mathcal {\tilde{H}}$ we can associate a Riemannian metric $g_\rho$ on M defined by $g_\rho (X,Y)=\frac{1}{2}(\omega _\rho (X,JY)+\omega _\rho (Y,JX))$. Let $\Sigma$ be a closed real surface and $f:\Sigma \rightarrow M$ be a smooth immersion. Define

$$\begin{aligned} \mathcal {A}(\rho )=\mathrm{Area}(f(\Sigma ),f^*(g_\rho ))=\int _\Sigma \mathrm{d}\mu _\rho , \end{aligned}$$

where $\mathrm{d}\mu _\rho$ is the volume form of the induced metric $f^*g_\rho$. We say that the area functional $\mathcal {A}$ has a critical point $\rho \in \mathcal {\tilde{H}}$ if for any $\phi (t)\in \mathcal {\tilde{H}}$ with $\phi (0)=\rho$, we have $\mathcal {A}'(0)=0$. Their first result, Corollary 1.2 of [1], says that if the area functional has a critical point, then the injective immersion is J-holomorphic. We say that $\rho \in \mathcal {\tilde{H}}$ is a stable point for the area functional $\mathcal {A}$ if $\mathcal {A}''(0)\ge 0$ for any $\phi (t)\in \mathcal {\tilde{H}}$, $\phi (0)=\rho$. Furthermore, if J is compatible with $\omega _\rho$, then we say that $\rho$ is a compatible stable point. For the stable case, their second theorem (Theorem 3.2 of [1]) says that, if the area functional has a compatible stable point, then the injective immersion is also J-holomorphic. The area functional is a natural candidate to be considered because for a J-holomorphic immersion, the area functional is constant in the same cohomology class (Proposition 2.2 of [1]) so that every point is both a critical point and a stable point for a J-holomorphic curve. Immediately following, Arezzo and Sun [2] generalized the results in [1] to arbitrary dimension and codimension as well as current case.

In [12], J. Sun considered a family of more general functionals defined in terms of the Kähler angle. In order to ensure $\omega _\rho =\omega +dd^c_J\rho$ is a (1, 1)-form, Sun considered the general functionals on a compact Kähler manifold. Let $(M,g,J,\omega )$ be a compact Kähler manifold. Recall that the Kähler angle $\alpha$ of a surface $\Sigma$ in M is defined by [3]

$$\begin{aligned} \omega |_{\Sigma }=\cos \alpha \mathrm{d}\mu _{\Sigma }, \end{aligned}$$

(1.1)

where $\mathrm{d}\mu _{\Sigma }$ is the induced volume form on $\Sigma$. We call an immersion $f:\Sigma \rightarrow M$ symplectic if $\cos \alpha >0$ and Lagrangian if $\cos \alpha \equiv 0$. As in [1], J. Sun fixed the immersion f and the complex structure J, and let the Kähler form vary in the fixed Kähler class. Then he defined a functional on $\mathcal {\tilde{H}}$ by

$$\begin{aligned} L_p(\rho )=\int _{\Sigma }\cos ^p\alpha _{\rho }\mathrm{d}\mu _{\rho }, \end{aligned}$$

(1.2)

where $\mathrm{d}\mu _{\rho }$ is the area form of the induced metric $f^* g_{\rho }$ on $\Sigma$, $\alpha _{\rho }$ is the Kähler angle of the immersion f with respect to the Kähler form $\omega _{\rho }$ and associated Riemannian metric $g_{\rho }$. When $p<0$ or p is not an integer, we assume the immersion to be symplectic in order to guarantee that the integral makes sense. When $p=0$, $L_0$ is just the area functional. J. Sun proved in [12] that if the functional $L_p$ has a critical point or a stable point in the fixed Kähler class, then the injective symplectic immersion is J-holomorphic.

In this paper, by considering the critical points and stable points of functional $L_p$, we investigate the existence of J-holomorphic curves in almost Hermitian manifolds. We prove that if the $L_p$ has a critical point or a stable point, then the immersion is J-holomorphic.

2 Definitions and preliminaries

Let M be a closed oriented smooth 2n-manifold. An almost complex structure on M is a differentiable endomorphism on the tangent bundle

$$\begin{aligned} J: TM\rightarrow TM\,\,\,\mathrm{with}\,\,\,J^2=-id. \end{aligned}$$

A manifold M with a fixed almost complex structure J is called an almost complex manifold denoted by (M, J). Suppose that M is an almost complex manifold with almost complex structure J, then for any $x\in M$, $T_x(M)\otimes _\mathbb {R}\mathbb {C}$ which is the complexification of $T_x(M)$, has the following decomposition (cf. [5]):

$$\begin{aligned} T_x(M)\otimes _\mathbb {R}\mathbb {C}=T^{1,0}_x+T^{0,1}_x, \end{aligned}$$

(2.3)

where $T^{1,0}_x$ and $T^{0,1}_x$ are the eigenspaces of J corresponding to the eigenvalues $\sqrt{-1}$ and $-\sqrt{-1}$, respectively. A complex tangent vector is of type (1, 0) (resp. (0, 1)) if it belongs to $T^{1,0}_x$ (resp. $T^{0,1}_x$). Let $T(M)\otimes _\mathbb {R}\mathbb {C}$ be the complexification of the tangent bundle. Similarly, let $T^*M\otimes _{\mathbb {R}}\mathbb {C}$ denote the complexification of the cotangent bundle $T^*M$. J can act on $T^*M\otimes _{\mathbb {R}}\mathbb {C}$ as follows:

$$\begin{aligned} \forall \alpha \in T^*M\otimes _{\mathbb {R}}\mathbb {C},\,\,\, J\alpha (\cdot )=-\alpha (J\cdot ). \end{aligned}$$

Hence $T^*M\otimes _{\mathbb {R}}\mathbb {C}$ has the following decomposition according to the eigenvalues $\pm \sqrt{-1}$:

$$\begin{aligned} T^*M\otimes _{\mathbb {R}}\mathbb {C}=\Lambda ^{1,0}_J\oplus \Lambda ^{0,1}_J. \end{aligned}$$

(2.4)

We can form exterior bundle $\Lambda ^{p,q}_J=\Lambda ^p\Lambda ^{1,0}_J\otimes \Lambda ^q\Lambda ^{0,1}_J$. Let $\Omega ^{p,q}_J(M)$ denote the space of $C^\infty$ sections of the bundle $\Lambda ^{p,q}_J$. Then we have a direct sum decomposition $\Omega ^k(M)=\bigoplus _{p+q=k}\Omega ^{p,q}_J(M)$. We denote the projections $\Omega ^k(M)\rightarrow \Omega ^{p,q}_J(M)$ by $\Pi ^{p,q}$. The exterior differential operator acts on $\Omega ^{p,q}_J$ as follows:

$$\begin{aligned} d\Omega ^{p,q}_J\subset \Omega ^{p-1,q+2}_J+\Omega ^{p+1,q}_J+\Omega ^{p,q+1}_J +\Omega ^{p+2,q-1}_J. \end{aligned}$$

(2.5)

Hence, d has the following decomposition:

$$\begin{aligned} d=A_J\oplus \partial _J\oplus \bar{\partial }_J\oplus \bar{A}_J, \end{aligned}$$

(2.6)

where $A_J\triangleq \Pi ^{p-1,q+2}\circ d$, $\partial _J\triangleq \Pi ^{p+1,q}\circ d$, $\bar{\partial }_J\triangleq \Pi ^{p,q+1}\circ d$ and $\bar{A}_J\triangleq \Pi ^{p+2,q-1}\circ d$. Let $\alpha$ be a (p, q)-form. We have following formulas (cf. [10, 11])

Proposition 2.1

$$\begin{aligned}&\partial _J\alpha (\xi _1,\cdot \cdot \cdot ,\xi _{p+1}, \bar{\eta }_1,\cdot \cdot \cdot ,\bar{\eta }_q)\\&\quad =\sum ^{p+1}_{k=1}(-1)^{k+1}\xi _k \alpha (\xi _1,\cdot \cdot \cdot ,\hat{\xi }_k,\cdot \cdot \cdot ,\bar{\eta }_q) \\&\quad \,\,\,\,\,\,+\sum _{1\le k<l\le p+1}(-1)^{k+1}\alpha ([\xi _k,\xi _l],\xi _1, \cdot \cdot \cdot ,\hat{\xi }_k,\cdot \cdot \cdot ,\hat{\xi }_l,\cdot \cdot \cdot ,\bar{\eta }_q) \\&\quad \,\,\,\,\,\,+ \sum _{1\le k\le p+1,1\le l\le q} (-1)^{k+l+p+1}\alpha ([\xi _k,\bar{\eta }_l],\xi _1, \cdot \cdot \cdot ,\hat{\xi }_k,\cdot \cdot \cdot ,\hat{\bar{\eta }}_l,\cdot \cdot \cdot , \bar{\eta }_q),\\&\quad \bar{\partial }_J\alpha (\xi _1,\cdot \cdot \cdot ,\xi _p,\bar{\eta }_1, \cdot \cdot \cdot ,\bar{\eta }_{q+1})\\&\quad =\sum ^{q+1}_{k=1}(-1)^{k+p+1}\bar{\eta }_k \alpha (\xi _1,\cdot \cdot \cdot ,\hat{\bar{\eta }}_k,\cdot \cdot \cdot ,\bar{\eta }_{q+1}) \\&\quad \,\,\,\,\,\,+\sum _{1\le k<l\le q+1}(-1)^{k+1}\alpha ([\bar{\eta }_k, \bar{\eta }_l],\xi _1,\cdot \cdot \cdot ,\hat{\bar{\eta }}_k,\cdot \cdot \cdot ,\hat{\bar{\eta }}_l,\cdot \cdot \cdot ,\bar{\eta }_{q+1}) \\&\quad \,\,\,\,\,\,+ \sum _{1\le k\le p,1\le l\le q+1} (-1)^{k+l+p}\alpha ([\xi _k,\bar{\eta }_l],\xi _1,\cdot \cdot \cdot , \hat{\xi }_k,\cdot \cdot \cdot ,\hat{\bar{\eta }}_l,\cdot \cdot \cdot ,\bar{\eta }_{q+1}),\\&A_J\alpha (\xi _1,\cdot \cdot \cdot ,\xi _{p-1},\bar{\eta }_1, \cdot \cdot \cdot ,\bar{\eta }_{q+2})\\&\quad =\sum _{1\le k<l\le q+2}(-1)^{k+l}\alpha ([\bar{\eta }_k, \bar{\eta }_l],\xi _1,\cdot \cdot \cdot ,\hat{\bar{\eta }}_k,\cdot \cdot \cdot , \hat{\bar{\eta }}_l,\cdot \cdot \cdot ,\bar{\eta }_{q+2}) \end{aligned}$$

and

$$\begin{aligned}&\bar{A}_J\alpha (\xi _1,\cdot \cdot \cdot ,\xi _{p+2},\bar{\eta }_1,\cdot \cdot \cdot ,\bar{\eta }_{q-1})\\&\quad =\sum _{1\le k<l\le p+2}(-1)^{k+l}\alpha ([\xi _k,\xi _l],\xi _1,\cdot \cdot \cdot ,\hat{\xi }_k,\cdot \cdot \cdot ,\hat{\xi }_l,\cdot \cdot \cdot ,\bar{\eta }_{q-1}), \end{aligned}$$

where $\xi _1,\cdot \cdot \cdot ,\xi _{p+2},\eta _1,\cdot \cdot \cdot ,\eta _{q+2}$ are vector fields of type (1, 0).

It is easy to see that $A_J$ and $\bar{A}_J$ are $\mathbb {R}$-linear operators of order 0. Recall that on an almost complex manifold (M, J), there exists Nijenhuis tensor $\mathcal {N}_J$ as follows:

$$\begin{aligned} 4\mathcal {N}_J=[JX,JY]-[X,Y]-J[X,JY]-J[JX,Y], \end{aligned}$$

(2.7)

where $X, Y\in TM$. By Newlander-Nirenberg Theorem (cf. [8]), $\mathcal {N}_J=0$ if and only if J is integrable, that is, J is a complex structure. Moreover, we have the following equivalent conditions (for details, see [5, 9]):

• 1. J is integrable;

• 2. $d=\partial _J\oplus \bar{\partial }_J$;

• 3. $\bar{\partial }^2_J=0$, $\partial ^2_J=0$;

• 4. If $\xi$ and $\eta$ are vector fields of type (1, 0), so is $[\xi ,\eta ]$.

Let (M, J) be an almost complex manifold and $(\Sigma ,j)$ be a Riemann surface. A smooth map $u:(\Sigma ,j)\rightarrow (M,J)$ is called a J-holomorphic curve if the differential $\mathrm{d}u$ is a complex linear map with respect to j and J:

$$\begin{aligned} J\circ \mathrm{d}u=\mathrm{d}u\circ j. \end{aligned}$$

(2.8)

Hence, $\bar{\partial }_Ju(X)=\frac{1}{2}[\mathrm{d}u(X)+J(u)\mathrm{d}u(jX)]=0$ if u is a J-holomorphic curve. By a result of Nijenhuis and Woolf (cf. [9]), the local J-holomorphic curves in an almost complex manifold are always exist.

Theorem 2.2

(cf. [9]) Let (M, J) be an almost complex manifold. Then to every point x of M and every complex tangent vector $v\in T(M)\otimes _\mathbb {R}\mathbb {C}$, there is a J-holomorphic curve passing through x with tangent vector v at x.

Suppose that (M, J) is a closed almost complex 2n-manifold. Let $\Sigma$ be a closed real surface and $f:\Sigma \rightarrow M$ be a smooth immersion. By (2.8), the definition of J-holomorphic curve, we can naturally give the definition of J-holomorphic immersion.

Definition 2.3

Let (M, J) is a closed almost complex 2n-manifold and $\Sigma$ be a closed real surface. We call a smooth immersion $f:\Sigma \rightarrow M$ J-holomorphic if $J_{f(p)}$ maps $f_{*p}(T_p\Sigma )$ onto itself for any point $p\in \Sigma$.

It is well known that there always exists complex structure j on surface $\Sigma$, that is, $(\Sigma ,j)$ is a closed Riemann surface. By the definition of J-holomorphic curve, if immersion $f:(\Sigma ,j)\rightarrow (M,J)$ is a J-holomorphic curve, then f is a J-holomorphic immersion. Conversely if $f:\Sigma \rightarrow M$ is a J-holomorphic immersion, then $f:(\Sigma ,f^*J)\rightarrow (M,J)$ is a J-holomorphic curve.

A symplectic structure on a differentiable manifold is a nondegenerate closed 2-form $\omega \in \Omega ^2$. A differentiable manifold with some fixed symplectic structure is called a symplectic manifold. Suppose $(M, \omega )$ is a closed symplectic manifold. An almost complex structure J is said to be tamed by $\omega$ when the bilinear form $\omega (\cdot ,J\cdot )$ is positive definite. The almost complex structure J is said to be compatible with $\omega$ when this same bilinear form is also symmetric, that is, $\omega (\cdot ,J\cdot )>0$ and $\omega (J\cdot ,J\cdot )=\omega (\cdot ,\cdot )$. We also call $\omega$ a J-compatible symplectic structure. It is well known that there always exists $\omega$-compatible almost complex structure J on $(M, \omega )$. Then we can define a J-invariant (J-compatible) Riemannian metric by $g(\cdot ,\cdot )\triangleq \omega (\cdot ,J\cdot )$. Such a quadruple $(M,g,J,\omega )$ is called an almost Kähler manifold. Recall that the energy of a smooth map $u:\Sigma \longrightarrow (M,g,J,\omega )$ is defined as the $L^2$-norm of the 1-form $\mathrm{d}u\in \Omega ^1(\Sigma , u^*TM)$:

$$\begin{aligned} E_J(u)\triangleq \frac{1}{2}\int _\Sigma |\mathrm{d}u|_J^2\mathrm{d}\mu _\Sigma . \end{aligned}$$

Here, the norm of the (real) linear map $L\triangleq \mathrm{d}u(z):T_z\Sigma \rightarrow T_{u(z)}M$ is defined by

$$\begin{aligned} |L|_J\triangleq |\xi |^{-1}\sqrt{|L(\xi )|^2_J+|L(j_\Sigma \xi )|^2_J} \end{aligned}$$

for $0\ne \xi \in T_z\Sigma$, where $|L(\xi )|^2_J=g(L(\xi ),L(\xi ))$. By Lemma 2.2.1 in [7],

$$\begin{aligned} E_J(u)=\int _\Sigma |\bar{\partial }_Ju|_J^2\mathrm{d}\mu _\Sigma +\int _\Sigma u^*\omega . \end{aligned}$$

Hence J-holomorphic curve $u:(\Sigma ,j)\longrightarrow (M,g,J,\omega )$ is a minimal surface with respect to the almost Kähler metric g. Under what condition a minimal surface is a J-holomorphic curve is an interesting question in differential geometry.

3 Critical point of $L_p$-functional

Suppose (M, J) is a closed almost complex 2n-manifold. One can construct a J-invariant Riemannian metric g on M. Such a metric g is called an almost Hermitian metric for (M, J). This then in turn gives a J-compatible nondegenerate 2-form F by $F(X,Y)=g(JX,Y)$, called the fundamental 2-form. Such a quadruple (M, g, J, F) is called a closed almost Hermitian manifold. If $dF=0$, then F will be written as $\omega$ and $(M,g,J,\omega )$ is called an almost Kähler manifold. By direct calculation, $F^n= n!\mathrm{d}\mu _g$, where $\mathrm{d}\mu _g$ is the volume form of M determined by g.

Proposition 3.1

(Wirtinger Inequality) (we refer to [4] for a direct and simple proof) Suppose that (M, g, J, F) is a closed almost Hermitian 2n-manifold. Let N be an oriented real smooth 2p-submanifold in M, and let $\mathrm{d}\mu _N$ be the Riemannian volume form on N associated with the metric $g|_N$. Set

$$\begin{aligned} \frac{1}{p!}F^p|_N=a\mathrm{d}\mu _N,\,\,\,a\in C^\infty (N). \end{aligned}$$

Then $|a|\le 1$ and the equality holds if and only if N is an almost complex submanifold of M.

Hence, we can define the Kähler angle $\alpha$ for a surface $\Sigma$ in almost Hermitian manifold (M, g, J, F) by

$$\begin{aligned} F|_{\Sigma }=\cos \alpha \mathrm{d}\mu _{\Sigma }. \end{aligned}$$

(3.9)

Note that a smooth map $u:\Sigma \longrightarrow (M,g,J,F)$ (an almost Hermitian manifold) is a J-holomorphic curve if and only if it is conformal with respect to g, i.e. its differential preserves angles or, equivalently, it preserves inner products up to a common positive factor. By Wirtinger Inequality and the definition of Kähler angle, we can easily get the following Proposition,

Proposition 3.2

Let (M, g, J, F) is a closed almost Hermitian 2n-manifold. Then $f:\Sigma \rightarrow (M,J)$ is a J-holomorphic immersion if and only if $\sin \alpha \equiv 0$.

Let (M, g, J, F) be an almost Hermitian 2n-manifold. After a simple calculation, we can get the following properties:

$$\begin{aligned}&d:\Omega ^0\longrightarrow \Omega ^1,\,\,\, d=\partial _J+\bar{\partial }_J. \end{aligned}$$

(3.10)

$$\begin{aligned}&A_J\circ \partial _J+\bar{\partial }^2_J+\bar{A}_J\circ \bar{\partial }_J+\partial ^2_J=0 : \Omega ^0\longrightarrow (\Omega ^{2,0}_J+\Omega ^{0,2}_J). \end{aligned}$$

(3.11)

$$\begin{aligned}&\partial _J\circ \bar{\partial }_J+\bar{\partial }_J\circ \partial _J=0: \Omega ^0\longrightarrow \Omega ^{1,1}. \end{aligned}$$

(3.12)

$$\begin{aligned}&d:\Omega ^1\longrightarrow \Omega ^2,\,\,\, d=A_J+\partial _J+\bar{\partial }_J+\bar{A}_J. \end{aligned}$$

(3.13)

By the above formulars, we get

Proposition 3.3

Let (M, g, J, F) be an almost Hermitian 2n-manifold. For any $\rho \in C^\infty (M,\mathbb {R})$, we have

$$\begin{aligned} dd^c_J\rho =2\sqrt{-1}\partial _J\bar{\partial }_J\rho +2\sqrt{-1}(\bar{A_J} \bar{\partial }_J\rho -A_J\partial _J\rho ). \end{aligned}$$

Proof

Firstly, by a simple calculation, we can get

$$\begin{aligned} dd^c_J\rho =2\sqrt{-1}\partial _J\bar{\partial }_J\rho +\sqrt{-1} (\bar{A_J}\bar{\partial }_J\rho -\partial ^2_J\rho )+\sqrt{-1} (\bar{\partial }^2_J\rho -A_J\partial _J\rho ). \end{aligned}$$

Since

$$\begin{aligned} d^2\rho= & {} d(\partial _J\rho +\bar{\partial }_J\rho ) \\= & {} \partial ^2_J\rho +A_J\partial _J\rho +\bar{\partial }_J\partial _J \rho +\bar{\partial }^2_J\rho +\bar{A_J}\bar{\partial }_J\rho +\partial _J \bar{\partial }_J\rho \\= & {} (\bar{\partial }_J\partial _J\rho +\partial _J\bar{\partial }_J\rho ) +(\partial ^2_J\rho +\bar{A_J}\bar{\partial }_J\rho )+(\bar{\partial }^2_J\rho +A_J\partial _J\rho ) \\= & {} 0, \end{aligned}$$

the corresponding individual components are equal to 0 respectively, that is, the (1, 1)-component $\bar{\partial }_J\partial _J\rho +\partial _J\bar{\partial }_J\rho =0$; the (2, 0)-component $\partial ^2_J\rho +\bar{A_J}\bar{\partial }_J\rho =0$; the (0, 2)-component $\bar{\partial }^2_J\rho +A_J\partial _J\rho =0$. Hence,

$$\begin{aligned} dd^c_J\rho =2\sqrt{-1}\partial _J\bar{\partial }_J\rho +2\sqrt{-1}(\bar{A_J} \bar{\partial }_J\rho -A_J\partial _J\rho ). \end{aligned}$$

$\square$

Let (M, g, J, F) be an almost Hermitian 2n-manifold. Let

$$\begin{aligned} \mathcal {H}\triangleq \{\rho \in C^\infty (M,\mathbb {R}):F_\rho \triangleq F+dd^c_J\rho \,\,\,\mathrm{tames}\,\,\,J\}, \end{aligned}$$

which is clearly a nonempty open subset of $C^\infty (M,\mathbb {R})$. Given $\rho \in {\mathcal H}$, define

$$\begin{aligned} F_{\rho }=F+dd^c_J\rho . \end{aligned}$$

(3.14)

In general, since J is not integrable, $dd^c_J\rho$ is not a (1, 1)-form. Thus, $F_{\rho }$ is not a J-compatible 2-form. The associated almost Hermitian metric is given by

$$\begin{aligned} g_\rho (X,Y)= & {} \frac{1}{2}(F_\rho (X,JY)+F_\rho (Y,JX)) \nonumber \\= & {} \Pi ^{1,1}(F_\rho )(X,JY)\nonumber \\= & {} (F+2\sqrt{-1}\partial _J\bar{\partial }_J\rho )(X,JY). \end{aligned}$$

(3.15)

Obviously, $F+2\sqrt{-1}\partial _J\bar{\partial }_J\rho$ is a J-compatible 2-form and $(g_\rho ,J,F+2\sqrt{-1}\partial _J\bar{\partial }_J\rho )$ is an almost Hermitian structure. Given the immersion $f:\Sigma \rightarrow M$, we have the induced metric and 2-form on $\Sigma$:

$$\begin{aligned} g'_{\rho }=f^*g_{\rho }, \ \ \ F'_{\rho }=f^*(F+2\sqrt{-1}\partial _J\bar{\partial }_J\rho ). \end{aligned}$$

(3.16)

The cosine of the Kähler angle $\alpha _{\rho }$ is define by

$$\begin{aligned} F'_{\rho }=\cos \alpha _{\rho } \mathrm{d}\mu _{g'_{\rho }}. \end{aligned}$$

(3.17)

Define the $L_p$-functional on $\mathcal {H}$ by

$$\begin{aligned} L_p(\rho )=\int _{\Sigma }\cos ^p\alpha _{\rho }\mathrm{d}\mu _{g'_{\rho }}. \end{aligned}$$

(3.18)

Definition 3.4

Given an immersion $F:\Sigma \rightarrow (M,g,J,F)$, we say that the functional $L_p$ has a critical point $\rho \in \mathcal {H}$ if for any $\varphi (t) \in \mathcal {H}$ with $\varphi (0) = \rho$

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}L_p(\varphi (t))=0. \end{aligned}$$

Stokes’ theorem immediately gives the following

Proposition 3.5

If $F:\Sigma \rightarrow (M,g,J,F)$ is a J-holomorphic immersion, then $L_p$ is constant on $\mathcal {H}$.

Proof

By Proposition 3.2, for each $\rho \in \mathcal {H}$, we have $\cos ^2\alpha _{\rho }\equiv 1$ on $\Sigma$. Without loss of generality, we may assume that $\cos \alpha _{\rho }\equiv 1$ on $\Sigma$ since $\cos \alpha _{\rho }$ is smooth on $\Sigma$. Then, $L_p(\rho )=\int _{\Sigma }\mathrm{d}\mu _{g'_{\rho }}$ is just the area functional $\mathcal {A}(\rho )$. By Proposition 2.3 in [1], we get that $L_p$ is constant on $\mathcal {H}$. $\square$

By the above proposition, we will find that if $F:\Sigma \rightarrow (M,g,J,F)$ is a J-holomorphic immersion, then every $\rho \in \mathcal {H}$ is the critical point of $L_p$. Our interest is in which sense the converse holds. Choose a $g'_0$-orthonormal basis $\{e_1,e_2\}$ of $T_p\Sigma$, then

$$\begin{aligned} \cos \alpha _0= F'_0(e_1,e_2) \end{aligned}$$

(3.19)

and

$$\begin{aligned} \cos \alpha _{\rho }=\frac{F'_{\rho }(e_1,e_2)}{\sqrt{\text{ det }(g'_{\rho }(e_i,e_j))}}. \end{aligned}$$

(3.20)

By (3.16),

$$\begin{aligned} F'_{\rho }=f^*(F+2\sqrt{-1}\partial _J\bar{\partial }_J\rho ) =F'_0+f^*(2\sqrt{-1}\partial _J\bar{\partial }_J\rho ), \end{aligned}$$

(3.21)

so that

$$\begin{aligned} F'_{\rho }(e_1,e_2)=\cos \alpha _0+(2\sqrt{-1}\partial _J \bar{\partial }_J\rho )(f_*e_1,f_*e_2). \end{aligned}$$

(3.22)

Hence, by (3.20), we have

$$\begin{aligned} \cos \alpha _{\rho }=\frac{\cos \alpha _0+(2\sqrt{-1}\partial _J \bar{\partial }_J\rho )(f_*e_1,f_*e_2)}{\sqrt{\text{ det }(g'_{\rho }(e_i,e_j))}}. \end{aligned}$$

(3.23)

Since $\{e_1,e_2\}$ is $g'_0$-orthonormal, by (3.15), we have

$$\begin{aligned} g'_{\rho }(e_i,e_j)= & {} g_{\rho }(f_*e_i,f_*e_j) \nonumber \\= & {} (F+2\sqrt{-1}\partial _J\bar{\partial }_J\rho )(f_*e_i,Jf_*e_j) \nonumber \\= & {} g(f_*e_i,f_*e_j)+(2\sqrt{-1}\partial _J\bar{\partial }_J\rho )(f_*e_i,Jf_*e_j) \nonumber \\= & {} \delta _{ij}+(2\sqrt{-1}\partial _J\bar{\partial }_J\rho )(f_*e_i,Jf_*e_j) . \end{aligned}$$

(3.24)

Therefore,

$$\begin{aligned} \text{ det }(g'_{\rho })= & {} 1+(2\sqrt{-1}\partial _J\bar{\partial }_J\rho )(f_*e_1,Jf_*e_1) +(2\sqrt{-1}\partial _J\bar{\partial }_J\rho )(f_*e_2,Jf_*e_2) \nonumber \\&+\,4(\sqrt{-1}\partial _J\bar{\partial }_J\rho )(f_*e_1,Jf_*e_1) \cdot (\sqrt{-1}\partial _J\bar{\partial }_J\rho )(f_*e_2,Jf_*e_2)\nonumber \\&-\,4[(\sqrt{-1}\partial _J\bar{\partial }_J\rho )(f_*e_1,Jf_*e_2)]^2 . \end{aligned}$$

(3.25)

Choose a g-orthonormal frame $\{e_1, e_2, \cdots , e_{2n}\}$ of $T_{f(p)}M$ such that $\{e_1, e_2\}$ spans the tangent space $T_p\Sigma$ and $\{e_3, \cdots , e_{2n}\}$ spans the normal space of $\Sigma$. Here, we identify $e_i$ with $f_*e_i$ for simplicity. Then the almost complex structure J takes the form

$$\begin{aligned} J=\left( \begin{array}{cccc} (J_1)_{4\times 4} &{}\quad 0_{4\times (2n-4)} \\ 0_{(2n-4)\times 4} &{}\quad (J_2)_{(2n-4)\times (2n-4)} \\ \end{array} \right) , \end{aligned}$$

(3.26)

where

$$\begin{aligned} J_1=\left( \begin{array}{cccc} 0 &{}\quad \cos \alpha _0 &{}\quad \sin \alpha _0 &{}\quad 0 \\ -\cos \alpha _0 &{}\quad 0 &{}\quad 0 &{}\quad -\sin \alpha _0 \\ -\sin \alpha _0 &{}\quad 0 &{}\quad 0 &{}\quad \cos \alpha _0 \\ 0 &{}\quad \sin \alpha _0 &{}\quad -\cos \alpha _0 &{}\quad 0 \\ \end{array} \right) , \end{aligned}$$

(3.27)

and $J_2$ satisfies $J_2^2=-Id_{2n-4}$.

In [1], Arezzo and Sun have gotten the following useful result

$$\begin{aligned} dd^c_J\rho (X,Y)= & {} -(\nabla ^2\rho )(X,JY)+(\nabla ^2\rho )(Y,JX) \nonumber \\&+\langle \nabla \rho ,(\nabla _YJ)X-(\nabla _XJ)Y\rangle _g, \end{aligned}$$

(3.28)

where $\nabla$ is the Levi-Civita connection of g. By Proposition 3.3, we have

$$\begin{aligned} 2\sqrt{-1}\partial _J\bar{\partial }_J\rho (X,Y)= & {} dd^c_J\rho (X,Y) -2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\rho -A_J\partial _J\rho )(X,Y) \nonumber \\= & {} -(\nabla ^2\rho )(X,JY)+(\nabla ^2\rho )(Y,JX) \nonumber \\&+\langle \nabla \rho ,(\nabla _YJ)X-(\nabla _XJ)Y\rangle _g \nonumber \\&-2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\rho -A_J\partial _J\rho )(X,Y) . \end{aligned}$$

(3.29)

Let $\varphi (t)$ be a variation coming from a 1-parameter deformation of $\varphi (0)=0$ in $\mathcal {H}$ with $\dot{\varphi }(0)=\gamma$. By (3.23), the $L_p$-functional has the following representation

$$\begin{aligned} L_p(\varphi (t))= & {} \int _{\Sigma }\cos ^p\alpha _{\varphi (t)}\mathrm{d}\mu _ {g'_{\varphi (t)}} \nonumber \\= & {} \int _{\Sigma }[\frac{\cos \alpha _0+(2\sqrt{-1}\partial _J\bar{\partial }_J\ \varphi (t))(f_*e_1,f_*e_2)}{\sqrt{\text{ det }(g'_{\varphi (t)}(e_i,e_j))}}]^p\mathrm{d}\mu _{g'_{\varphi (t)}}. \end{aligned}$$

(3.30)

In the following part, we will compute the first variation of the $L_p$-functional. By (3.25), we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\text{ det }(g'_{\varphi (t)})=(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_1,Je_1)+(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_2,Je_2). \end{aligned}$$

(3.31)

Then, by (3.23) and the above formualr, we have

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\cos \alpha _{\varphi (t)}\nonumber \\&\quad = (2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_1,e_2)-\frac{1}{2} \cos \alpha _0\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\text{ det }(g'_{\varphi (t)})\nonumber \\&\quad = (2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_1,e_2) -\frac{1}{2}\cos \alpha _0[(2\sqrt{-1}\partial _J \bar{\partial }_J\gamma )(e_1,Je_1)\nonumber \\&\qquad +(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_2,Je_2)]. \end{aligned}$$

(3.32)

With (3.26) and (3.29), by a direct computation,

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\cos \alpha _{\varphi (t)}\nonumber \\= & {} \cos \alpha _0[(\nabla ^2\gamma )(e_1,e_1)+(\nabla ^2\gamma )(e_2,e_2)] +\sin \alpha _0[(\nabla ^2\gamma )(e_1,e_4)+(\nabla ^2\gamma )(e_2,e_3)]\nonumber \\&+\langle \nabla \gamma ,(\nabla _{e_2}J)e_1-(\nabla _{e_1}J)e_2\rangle _g -2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma )(e_1,e_2) \nonumber \\&-\frac{1}{2}\cos \alpha _0\left\{ (1+\cos ^2\alpha _0)(\nabla ^2\gamma )(e_1,e_1)+(1+\cos ^2\alpha _0)(\nabla ^2\gamma )(e_2,e_2) \right. \nonumber \\&\left. +2\sin \alpha _0\cos \alpha _0(\nabla ^2\gamma )(e_2,e_3)+\sin ^2\alpha _0(\nabla ^2\gamma )(e_3,e_3)\right. \nonumber \\&\left. +2\sin \alpha _0\cos \alpha _0(\nabla ^2\gamma )(e_1,e_4)+\sin ^2\alpha _0(\nabla ^2\gamma )(e_4,e_4)\right\} \nonumber \\&-\frac{1}{2}\cos \alpha _0\{\langle \nabla \gamma ,(\nabla _{Je_1}J)e_1-(\nabla _{e_1}J)Je_1\rangle _g -2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma )(e_1,Je_1) \}\nonumber \\&-\frac{1}{2}\cos \alpha _0\{\langle \nabla \gamma ,(\nabla _{Je_2}J)e_2-(\nabla _{e_2}J)Je_2\rangle _g -2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma )(e_2,Je_2) \}\nonumber \\= & {} \frac{1}{2}\sin ^2\alpha _0 \left\{ \cos \alpha _0\left[ (\nabla ^2\gamma )(e_1,e_1)+(\nabla ^2\gamma )(e_2,e_2) -(\nabla ^2\gamma )(e_3,e_3)-(\nabla ^2\gamma )(e_4,e_4)\right] \right. \nonumber \\&\left. +2\sin \alpha _0\left[ (\nabla ^2\gamma )(e_1,e_4)+(\nabla ^2\gamma )(e_2,e_3)\right] \right\} \nonumber \\&+\langle \nabla \gamma ,(\nabla _{e_2}J)e_1-(\nabla _{e_1}J)e_2\rangle _g -2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma )(e_1,e_2) \nonumber \\&-\frac{1}{2}\cos \alpha _0\{\langle \nabla \gamma ,(\nabla _{Je_1}J)e_1-(\nabla _{e_1}J)Je_1\rangle _g -2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma )(e_1,Je_1) \}\nonumber \\&-\frac{1}{2}\cos \alpha _0\{\langle \nabla \gamma ,(\nabla _{Je_2}J)e_2-(\nabla _{e_2}J)Je_2\rangle _g -2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma )(e_2,Je_2) \}. \nonumber \\&\end{aligned}$$

(3.33)

Lemma 3.6

Let (M, g, J, F) be an almost Hermitian manifold and $f:\Sigma \rightarrow M$ be an injective immersion such that $\cos \alpha _0>0$. Set $d:M\rightarrow \mathbb {R}$ any smooth extension from a tubular neighborhood of $f(\Sigma )$ to M of the distance function from $f(\Sigma )$, i.e. $d(q) = dist(q, f(\Sigma ))$ for q sufficiently near $f(\Sigma )$. If

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0} L_p \left( F_{\rho } + tdd^c_J\left( \frac{d^2}{2}\right) \right) = 0 \end{aligned}$$

for some $p\in \mathbb {Z}-\{1\}$ and $\rho \in \mathcal {H}$, then the immersion is J-holomorphic.

Proof

Without loss of generality, we assume that $\rho \equiv 0$ so that $F_{\rho }=F$. Let $\varphi (t)$ be any curve in ${\mathcal H}$ such that $\varphi (0)=\rho \equiv 0$ and $\dot{\varphi }(0)=\gamma$. Fix a point $p\in \Sigma$ and take an orthonormal basis $\{e_1,e_2\}$ of $T_p\Sigma$ so that the complex structure J takes the form (3.26). By (3.24) and (3.31), it is easy to see that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\mathrm{d}\mu _{g'_{\varphi (t)}}=\frac{1}{2} \sum _{i=1}^{2}(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_i,Je_i)\mathrm{d}\mu _{g'_0}. \end{aligned}$$

Then, with (3.26) and (3.29), we obtain that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\mathrm{d}\mu _{g'_{\varphi (t)}}= & {} \frac{1}{2}\left\{ (1+\cos ^2\alpha _0)(\nabla ^2\gamma ) (e_1,e_1)+(1+\cos ^2\alpha _0)(\nabla ^2\gamma )(e_2,e_2) \right. \nonumber \\&\left. +2\sin \alpha _0\cos \alpha _0(\nabla ^2\gamma )(e_2,e_3)+\sin ^2\alpha _0 (\nabla ^2\gamma )(e_3,e_3)\right. \nonumber \\&\left. +2\sin \alpha _0\cos \alpha _0(\nabla ^2\gamma )(e_1,e_4)+\sin ^2\alpha _0 (\nabla ^2\gamma )(e_4,e_4)\right. \nonumber \\&+\langle \nabla \gamma ,(\nabla _{Je_1}J)e_1-(\nabla _{e_1}J)Je_1+(\nabla _{Je_2}J) e_2-(\nabla _{e_2}J)Je_2\rangle _g\nonumber \\&-2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma ) (e_1,Je_1)\nonumber \\&-2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma ) (e_2,Je_2) \}\mathrm{d}\mu _{g'_0}. \end{aligned}$$

(3.34)

By (3.33) and (3.34),

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}L_p(\varphi (t))\nonumber \\&\quad = p\int _{\Sigma }\cos ^{p-1}\alpha _0\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\cos \alpha _{\varphi (t)}\mathrm{d}\mu _{g'_0} +\int _{\Sigma }\cos ^p\alpha _0\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\mathrm{d}\mu _{g'_{\varphi (t)}}\nonumber \\&\quad = \int _{\Sigma }\frac{p\cos ^{p-1}\alpha _0\sin ^2\alpha _0}{2} \left\{ \cos \alpha _0\left[ (\nabla ^2\gamma )(e_1,e_1) +(\nabla ^2\gamma )(e_2,e_2)\right. \right. \nonumber \\&\qquad \left. \left. -(\nabla ^2\gamma )(e_3,e_3)-(\nabla ^2\gamma )(e_4,e_4)\right] \right. \nonumber \\&\qquad \left. +2\sin \alpha _0\left[ (\nabla ^2\gamma ) (e_1,e_4)+(\nabla ^2\gamma )(e_2,e_3)\right] \right\} \mathrm{d}\mu _{g'_0}\nonumber \\&\qquad +\int _{\Sigma }\frac{\cos ^p\alpha _0}{2}\left\{ (1+\cos ^2\alpha _0) \left[ (\nabla ^2\gamma )(e_1,e_1)+(\nabla ^2\gamma )(e_2,e_2)\right] \right. \nonumber \\&\qquad \left. +\sin ^2\alpha _0\left[ (\nabla ^2\gamma )(e_3,e_3) +(\nabla ^2\gamma )(e_4,e_4)\right] \right. \nonumber \\&\qquad \left. +2\sin \alpha _0\cos \alpha _0\left[ (\nabla ^2\gamma )(e_2,e_3)+ (\nabla ^2\gamma )(e_1,e_4)\right] \right\} \mathrm{d}\mu _{g'_0}+\Phi \nonumber \\&\quad = \int _{\Sigma }\{\frac{1}{2}\cos ^p\alpha _0 (1+\cos ^2\alpha _0+p\sin ^2\alpha _0) [(\nabla ^2\alpha _0)(e_1,e_1)+(\nabla ^2\alpha _0)(e_2,e_2)]\nonumber \\&\qquad +\frac{1-p}{2}\sin ^2\alpha _0\cos ^p\alpha _0[(\nabla ^2\alpha _0)(e_3,e_3) +(\nabla ^2\alpha _0)(e_4,e_4)]\nonumber \\&\qquad +\sin \alpha _0\cos ^{p-1}\alpha _0(\cos ^2\alpha _0+p\sin ^2\alpha _0)(\nabla ^2\alpha _0) (e_2,e_3) \nonumber \\&\qquad +\sin \alpha _0\cos ^{p-1}\alpha _0(\cos ^2\alpha _0+p\sin ^2\alpha _0) (\nabla ^2\alpha _0)(e_1,e_4)\}\mathrm{d}\mu _{g'_0}+\Phi , \end{aligned}$$

(3.35)

where

$$\begin{aligned} \Phi= & {} p\int _{\Sigma }\cos ^{p-1}\alpha _0\langle \nabla \gamma ,(\nabla _{e_2}J) e_1-(\nabla _{e_1}J)e_2\rangle _g\mathrm{d}\mu _{g'_0}\\&- p\int _{\Sigma }\cos ^{p-1}\alpha _02\sqrt{-1}(\bar{A_J}\bar{\partial }_J \gamma -A_J\partial _J\gamma )(e_1,e_2)\mathrm{d}\mu _{g'_0}\\&- \frac{p}{2}\int _{\Sigma }\cos ^p\alpha _0\{\langle \nabla \gamma , (\nabla _{Je_1}J)e_1-(\nabla _{e_1}J)Je_1\rangle _g\\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, -2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma )(e_1,Je_1) \}\mathrm{d}\mu _{g'_0}\nonumber \\&- \frac{p}{2}\int _{\Sigma }\cos ^p\alpha _0\{\langle \nabla \gamma , (\nabla _{Je_2}J)e_2-(\nabla _{e_2}J)Je_2\rangle _g\\&-2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J\partial _J\gamma )(e_2,Je_2) \}\mathrm{d}\mu _{g'_0}\nonumber \\&+\int _{\Sigma }\frac{\cos ^p\alpha _0}{2}\langle \nabla \gamma , (\nabla _{Je_1}J)e_1 -(\nabla _{e_1}J)Je_1+(\nabla _{Je_2}J)e_2-(\nabla _{e_2}J)Je_2 \rangle _g\mathrm{d}\mu _{g'_0}\\&-\int _{\Sigma }\cos ^p\alpha _0\sqrt{-1}(\bar{A_J}\bar{\partial }_J \gamma -A_J\partial _J\gamma )(e_1,Je_1) \mathrm{d}\mu _{g'_0}\\&-\int _{\Sigma }\cos ^p\alpha _0\sqrt{-1}(\bar{A_J}\bar{\partial }_J\gamma -A_J \partial _J\gamma )(e_2,Je_2)\mathrm{d}\mu _{g'_0}. \end{aligned}$$

We identify $\Sigma$ with its image in M. Denote d the distance function of M from $\Sigma$ with respect to the metric g, that is, for $q\in M$, $d(q)=dist_g(q,\Sigma )$. Recall that $\xi =\frac{1}{2}d^2$ is smooth in a neighborhood of $\Sigma$ in M (cf. [6]). By Proposition 2.6 in [1], for any $p\in \Sigma$, the hessian $Hess(\xi )(p)$ represents the orthogonal projection on the normal space to $\Sigma$ at p, that is, for each $X,Y\in T_{p}M$, we have

$$\begin{aligned} \nabla ^2 (\xi )(X,Y)(p)=\langle X^{\perp },Y^{\perp }\rangle , \end{aligned}$$

(3.36)

where $T_{p}M=T_{p}\Sigma \oplus N_{p}\Sigma$ and $X^{\perp }$ is the projection of X onto $N_{p}\Sigma$. Next, we will take special test function $\gamma$ to be a smooth function on M such that $\gamma =\frac{1}{2}d^2$ in a neighborhood of $\Sigma$ in M. Since $\{e_1,e_2\}$ is an orthonormal basis of $T_p\Sigma$, it is easy to see that $e_1^{\perp }=0$ and $e_2^{\perp }=0$. Hence,

$$\begin{aligned}&\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}L_p(\varphi (t))\nonumber \\&\quad = \int _{\Sigma } \frac{1-p}{2}\sin ^2\alpha _0\cos ^p\alpha _0[(\nabla ^2\alpha _0) (e_3,e_3)+(\nabla ^2\alpha _0)(e_4,e_4)]\mathrm{d}\mu _{g'_0}+\Phi .\qquad \end{aligned}$$

(3.37)

It is well known that both $\nabla$ and $\bar{A_J}\bar{\partial }_J-A_J\partial _J$ are $\mathbb {R}$-linear operators of order 1. So by the choice of $\gamma =\frac{1}{2}d^2$ and the definition of d, we can easily get $\Phi =\int _{\Sigma }(\cdot )\mathrm{d}\mu _{g'_0}=0$. Then by (3.36) and (3.37), we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0} L_p (F+tdd^c_J(\frac{d^2}{2}))=(1-p)\int _{\Sigma }\sin ^2\alpha _0\cos ^p\alpha _0d \mu _{g'_0}=0. \end{aligned}$$

(3.38)

On the other hand, by our assumption, $\cos \alpha _0>0$, $p\ne 1$ and $\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}L_p(F+t2\sqrt{-1}\partial _J\bar{\partial }_J(\frac{d^2}{2}))=0$. Therefore, we must have $\sin \alpha _0\equiv 0$. By Proposition 3.2, to prove the theorem, it suffices to show that $\sin \alpha _0\equiv 0$ on $\Sigma$. Hence, this completes the proof of Lemma 3.6.

Theorem 3.7

Let (M, g, J, F) be an almost Hermitian manifold and $f:\Sigma \rightarrow M$ be an injective immersion such that $\cos \alpha _0>0$. If for some $p\in \mathbb {Z}-\{1\}$, the functional $L_p$ has a critical point in $\mathcal {H}$, then the immersion is J-holomorphic.

If $dF=0$, then F will be written as $\omega$ and $(M,g,J,\omega )$ is called an almost Kähler manifold. The condition $\cos \alpha _0>0$ is just show that $f:\Sigma \rightarrow M$ is an injective symplectic immersion. Then Theorem 3.7 can be expressed as,

Corollary 3.8

Let $(M,g,J,\omega )$ be an almost Kähler manifold and $f:\Sigma \rightarrow M$ be an injective symplectic immersion. If for some $p\in \mathbb {Z}-\{1\}$, the functional $L_p$ has a critical point in $\mathcal {H}$, then the immersion is J-holomorphic.

When $p=0$, $L_0(\rho )$ is just the area functional $\mathcal {A}(\rho )$. Then the integrand of the right hand side of (3.38) becomes $\sin ^2\alpha _0$. Hence, we get

Corollary 3.9

(see [1]) Let (M, g, J, F) be an almost Hermitian manifold and $f:\Sigma \rightarrow M$ be an injective immersion. If the functional $L_0$ has a critical point in $\mathcal {H}$, then the immersion is J-holomorphic.

Let $(M,g,J,\omega )$ be an almost Kähler manifold and $f:\Sigma \rightarrow M$ be an injective immersion. Suppose that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0} L_p \left( \omega _{\rho } + tdd^c_J\left( \frac{d^2}{2}\right) \right) = 0 \end{aligned}$$

for some $p\in 2\mathbb {Z}^+$ and $\rho \in \mathcal {H}$. By (3.38), we have

$$\begin{aligned} \sin ^2\alpha _0\cos ^p\alpha _0 \equiv 0 \end{aligned}$$

on $\Sigma$. Then we will obtain $\sin \alpha _0\equiv 0$ or $\cos \alpha _0\equiv 0$ on $\Sigma$. If $\sin \alpha _0\equiv 0$, the immersion is J-holomorphic. If $\cos \alpha _0\equiv 0$, the immersion is Lagrangian.

4 Stable point of $L_p$-functional

In light of our knowledge about the relationship between stable minimal surfaces and holomorphic curves, it is natural to look at special properties of the second variation of the functional $L_p$.

Definition 4.1

Given a symplectic immersion $F:\Sigma ^2\rightarrow (M,\bar{\omega },J_M,\bar{g})$, we say that $\rho \in \mathcal {H}_p$ is a stable point for the functional $L_p$ if for any $\varphi (t)\in \mathcal {H}$ with $\varphi (0)=\rho$

$$\begin{aligned} \frac{d^2}{dt^2}|_{t=0}L_p(\varphi (t))\ge 0. \end{aligned}$$

Let (M, g, J, F) be an almost Hermitian manifold and $f:\Sigma \rightarrow M$ be an injective immersion such that $\cos \alpha _0>0$ as in the previous section. Take any curve $\varphi (t)\in \mathcal {H}$ with $\varphi (0)=0$, $\dot{\varphi }(0)=\gamma$ and $\ddot{\varphi }(0)=\zeta$. By (3.23),

$$\begin{aligned} L_p(\varphi (t))=\int _{\Sigma }[\cos \alpha _0+(2\sqrt{-1}\partial _J \bar{\partial }_J\varphi )(e_1,e_2)]^p\text{ det }(g'_{\varphi })^{\frac{1-p}{2}}. \end{aligned}$$

Then, we have

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}L_p(\varphi (t))= & {} \int _{\Sigma }p[\cos \alpha _0+(2\sqrt{-1} \partial _J\bar{\partial }_J\varphi )]^{p-1}\text{ det }(g'_{\varphi })^{\frac{1-p}{2}} \frac{\mathrm{d}}{\mathrm{d}t}(2\sqrt{-1}\partial _J\bar{\partial }_J\varphi )\\&+\frac{1-p}{2}[\cos \alpha _0+(2\sqrt{-1}\partial _J\bar{\partial }_J\varphi )]^p \text{ det }(g'_{\varphi })^{\frac{-1-p}{2}}\frac{\mathrm{d}}{\mathrm{d}t}\text{ det }(g'_{\varphi }) . \end{aligned}$$

Hence, combined the above formula, (3.25) and (3.31), the second variation formula for the functional $L_p$ is given by

$$\begin{aligned}&\frac{d^2}{dt^2}|_{t=0}L_p(\varphi (t)) \nonumber \\&\quad =\int _{\Sigma } p(p-1)\cos ^{p-2}\alpha _0[\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}(2\sqrt{-1}\partial _J \bar{\partial }_J\varphi (t))(e_1,e_2)]^2 \nonumber \\&\qquad +\frac{p(1-p)}{2}\cos ^{p-1}\alpha _0\frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\text{ det }(g'_{\varphi (t)}) \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}(2\sqrt{-1}\partial _J\bar{\partial }_J\varphi (t))(e_1,e_2)\nonumber \\&\qquad +p\cos ^{p-1}\alpha _0\frac{d^2}{dt^2}|_{t=0}(2\sqrt{-1}\partial _J \bar{\partial }_J\varphi (t))(e_1,e_2)\nonumber \\&\qquad +\frac{p(1-p)}{2}\cos ^{p-1}\alpha _0 \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}(2\sqrt{-1}\partial _J\bar{\partial }_J\varphi (t))(e_1,e_2) \frac{\mathrm{d}}{\mathrm{d}t}|_{t=0}\text{ det }(g'_{\varphi (t)})\nonumber \\&\qquad +\frac{(p-1)(p+1)}{4}\cos ^p\alpha _0[\frac{d}{dt}|_{t=0}\text{ det }(g'_{\varphi (t)})]^2 +\frac{1-p}{2}\cos ^p\alpha _0\frac{d^2}{dt^2}|_{t=0} \text{ det }(g'_{\varphi (t)})\nonumber \\&\quad = \int _{\Sigma }p(p-1)\cos ^{p-2}\alpha _0[(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma ) (e_1,e_2)]^2\nonumber \\&\qquad +p(1-p)\cos ^{p-1}\alpha _0(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_1,e_2) \nonumber \\&\qquad \quad \cdot [(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_1,Je_1)+(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_2,Je_2)]\nonumber \\&\qquad +p\cos ^{p-1}\alpha _0(2\sqrt{-1}\partial _J\bar{\partial }_J\zeta )(e_1,e_2)\nonumber \\&\qquad +\frac{(p-1)(p+1)}{4}\cos ^p\alpha _0 [(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma ) (e_1,Je_1)+(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_2,Je_2)]^2\nonumber \\&\qquad +\frac{1-p}{2}\cos ^p\alpha _0 \{(2\sqrt{-1}\partial _J\bar{\partial }_J\zeta ) (e_1,Je_1)+(2\sqrt{-1}\partial _J\bar{\partial }_J\zeta )(e_2,Je_2)\}\nonumber \\&\qquad +\frac{1-p}{2}\cos ^p\alpha _0\{ 2(2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_1,Je_1) (2\sqrt{-1}\partial _J\bar{\partial }_J\gamma )(e_2,Je_2)\nonumber \\&\qquad -2[(2\sqrt{-1}\partial \bar{\partial }\gamma )(e_1,Je_2)]^2\}. \end{aligned}$$

(4.39)

Lemma 4.2

Let (M, g, J, F) be an almost Hermitian manifold and $f:\Sigma \rightarrow M$ be an injective immersion such that $\cos \alpha _0>0$ as above. Set $d:M\rightarrow \mathbb {R}$ any smooth extension from a tubular neighborhood of $f(\Sigma )$ to M of the distance function from $f(\Sigma )$, i.e. $d(q) = dist(q, f(\Sigma ))$ for q sufficiently near $f(\Sigma )$. If

$$\begin{aligned} \frac{d^2}{dt^2}|_{t=0}L_p\left( F_{\rho }+\frac{t^2}{2}dd^c_J\left( \frac{d^2}{2}\right) \right) =0 \end{aligned}$$

for some $p\in \mathbb {Z}-\{1\}$ and $\rho \in \mathcal {H}$, then the immersion is J-holomorphic.

Proof

Without loss of generality, we assume that $\rho =0$. Moreover we take $\varphi (t)=\frac{t^2}{2}\zeta$ so that $\gamma =0$. Then formula (4.39) becomes

$$\begin{aligned} \frac{d^2}{dt^2}|_{t=0}L_p(\varphi (t))= & {} \int _{\Sigma }p\cos ^{p-1}\alpha _0(2\sqrt{-1}\partial _J \bar{\partial }_J\zeta )(e_1,e_2)\nonumber \\&+\frac{1-p}{2}\cos ^p\alpha _0(2\sqrt{-1}\partial _J\bar{\partial }_J\zeta ) (e_1,Je_1)\nonumber \\&+\frac{1-p}{2}\cos ^p\alpha _0(2\sqrt{-1}\partial _J\bar{\partial }_J\zeta )(e_2,Je_2). \end{aligned}$$

(4.40)

By (3.26), (3.27) and (3.29), we have

$$\begin{aligned} 2\sqrt{-1}\partial _J\bar{\partial }_J\zeta (e_1,e_2)= & {} -(\nabla ^2\zeta )(e_1,Je_2)+(\nabla ^2\zeta )(e_2,Je_1) \nonumber \\&+\langle \nabla \zeta ,(\nabla _{e_2}J)e_1-(\nabla _{e_1}J)e_2\rangle _g \nonumber \\&-2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\zeta -A_J\partial _J\zeta )(e_1,e_2) \nonumber \\= & {} \cos \alpha _0[(\nabla ^2\zeta )(e_1,e_1)+(\nabla ^2\zeta )(e_2,e_2)]\nonumber \\&+\sin \alpha _0[(\nabla ^2\zeta )(e_1,e_4)+(\nabla ^2\zeta )(e_2,e_3)] \nonumber \\&+\langle \nabla \zeta ,(\nabla _{e_2}J)e_1-(\nabla _{e_1}J)e_2\rangle _g \nonumber \\&-2\sqrt{-1}(\bar{A_J}\bar{\partial }_J\zeta -A_J\partial _J\zeta )(e_1,e_2). \end{aligned}$$

(4.41)

Now, we take $\zeta$ to be a smooth function on M so that $\zeta =\frac{d^2}{2}$ in a neighborhood of $\Sigma$. Then by (3.36) and the fact that both $\nabla$ and $\bar{A_J}\bar{\partial }_J-A_J\partial _J$ are $\mathbb {R}$-linear operators of order 1, we have $2\sqrt{-1}\partial _J\bar{\partial }_J\zeta (e_1,e_2)=0$ when restricting on $\Sigma$. Similarly, we have

$$\begin{aligned} (2\sqrt{-1}\partial _J\bar{\partial }_J\zeta )(e_1,Je_1)=(2\sqrt{-1}\partial _J \bar{\partial }_J\zeta )(e_2,Je_2)=\sin ^2\alpha _0. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \frac{d^2}{dt^2}|_{t=0}L_p\left( F_{\rho }+\frac{t^2}{2}dd^c_J\left( \frac{d^2}{2}\right) \right) =(1-p)\int _{\Sigma }\cos ^p\alpha _0\sin ^2\alpha _0=0. \end{aligned}$$

Then, we obtain $\sin \alpha _0=0$ since we have assumed that $\cos \alpha _0>0$ and $p\ne 1$. By Proposition 3.2, this proves the lemma.

If $L_p$ has a stable point $\rho =0$, then by Definition 4.1, we have

$$\begin{aligned} \frac{d^2}{dt^2}|_{t=0}L_p\left( F_{\rho }+\frac{t^2}{2}dd^c_J(\zeta )\right) \ge 0. \end{aligned}$$

(4.42)

It easy to see that

$$\begin{aligned} \frac{d^2}{dt^2}|_{t=0}L_p\left( F_{\rho }-\frac{t^2}{2}dd^c_J(\zeta )\right) =-\frac{d^2}{dt^2}|_{t=0}L_p\left( F_{\rho }+\frac{t^2}{2}dd^c_J(\zeta )\right) . \end{aligned}$$

Replacing $\zeta$ by $-\zeta$ in (4.42), we can get $-\frac{d^2}{dt^2}|_{t=0}L_p(F_{\rho }+\frac{t^2}{2}dd^c_J(\zeta ))\ge 0$. That means

$$\begin{aligned} \frac{d^2}{dt^2}|_{t=0}L_p\left( F_{\rho }+\frac{t^2}{2}dd^c_J(\zeta )\right) =0. \end{aligned}$$

Then, with Lemma 4.2, we can easily get the following theorem

Theorem 4.3

Let (M, g, J, F) be an almost Hermitian manifold and $f:\Sigma \rightarrow M$ be an injective immersion such that $\cos \alpha _0>0$. If the functional $L_p$ ($p\in \mathbb {Z}-\{1\}$) has a stable point in $\mathcal {H}$, then the immersion is J-holomorphic.