On a generalized Calabi-Yau equation

TL;DR

This paper extends a non-existence result for the generalized Calabi-Yau equation, originally proved in complex dimension 2, to manifolds of arbitrary dimension, advancing understanding in almost-Kähler geometry.

Contribution

It generalizes a key non-existence theorem for the generalized Calabi-Yau equation beyond complex dimension 2 to all dimensions.

Findings

Non-existence result extended to arbitrary dimensions.

Supports the conjecture about solutions in almost-Kähler manifolds.

Provides new insights into the structure of the generalized Calabi-Yau equation.

Abstract

Dealing with the generalized Calabi-Yau equation proposed by Gromov on closed almost-K\"ahler manifolds, we extend to arbitrary dimension a non-existence result proved in complex dimension 2.

Equations

$$(\omega +\sqrt{-1}\partial \bar{\partial }\varphi {)}^n=e^F{\omega }^n,$$

$$\omega +\sqrt{-1}\partial \bar{\partial }\varphi >0,$$

$${\int }_M{e}^F{\omega }^n={\int }_M{\omega }^n.$$

$${\omega }^{\prime n}=\sigma .$$

$$f:M\to M,$$

$$f^{\ast }{\omega }^n=\sigma .$$

$${\omega }^{\prime }=\omega (\varphi )\equiv \omega +d(Jd\varphi ),$$

$$\varphi \in C^{\infty }(M;\mathbb{R}).$$

$$(Jd\varphi )(X)=d\varphi (JX),$$

$$\varphi \in C^{\infty }(M;\mathbb{R}),$$

$$\{\begin{array}{l}{\omega }^{\prime n}=\sigma ,\\ {\omega }^{\prime }=\omega (\varphi )\equiv \omega +dJd\varphi \mathrm{tames}J,\end{array}$$

$$d(Jd\varphi )$$

$$=\sqrt{-1}(\partial +\bar{\partial })(\partial -\bar{\partial })\varphi$$

$$=2\sqrt{-1}\bar{\partial }\partial \varphi .$$

$$\varphi$$

$$F(\varphi ){\omega }^n=(\omega (\varphi ){)}^n.$$

$${\int }_M{\omega }^n={\int }_{M}f{\omega }^n.$$

$$\{\begin{array}{l}F(\varphi )=f,\\ \omega (\varphi )\equiv \omega +dJd\varphi \mathrm{tames}J.\end{array}$$

$$A$$

$$B$$

$$A_{+}$$

$$B_{+}$$

$$F(A_{+})\subset B_{+}.$$

$$F{|}_{A_{+}}:A_{+}\to B_{+},$$

$${\int }_M\sigma ={\int }_M{\omega }^2.$$

$${\tilde{\omega }}^2=\sigma .$$

$$J(X+iY)=JX+iJY$$

$$J^2=-Id$$

$$TM\otimes \mathbb{C}=T^{1,0}\oplus T^{0,1},$$

$$T^{1,0}=\{X\in TM\otimes \mathbb{C}:JX=iX\}.$$

$${\wedge }^2{T}^{\ast }M\otimes \mathbb{C}=T_{2,0}+T_{1,1}+T_{0,2},$$

$$\tau (\varphi )=P_{2,0}(\omega (\varphi )),$$

$$H(\varphi )=P_{1,1}(\omega (\varphi )).$$

Full text

On a generalized Calabi-Yau equation††thanks: Supported by

NSF Grant (China) 10671171.

Hongyu WANG     Peng ZHU

School of Mathematical Science, Yangzhou University,

Yangzhou, Jiangsu 225002, P. R. China. E-mail: hywang@yzu.edu.cnE-mail: zhupeng2004@yahoo.com.cn

Abstract. Dealing with the generalized Calabi-Yau equation proposed by Gromov on closed almost-Kähler manifolds, we extend to arbitrary dimension a non-existence result proved in complex dimension $2$.

AMS classification: 53C07; 53D05; 58J99.

Keywords: Calabi-Yau equation, symplectic form, almost complex structure, Hermitian metric, Nijenhuis tensor, pseudo holomorphic function

1 Introduction

The Calabi conjecture [3] asserts that any representative of the first Chern class of a closed Kähler manifold $(M,\omega)$ of real dimension $2n$ can be written as the Ricci curvature of a Kähler metric $\omega^{\prime}$ cohomologous to $\omega$. This conjecture was proved by S. T. Yau [16]. Yau’s result is equivalent to finding a Kähler metric in a given Kähler class with a prescribed volume form. More precisely, by $\partial\bar{\partial}$-lemma on a Kähler manifold, this amounts to solve the complex Monge-Ampère equation:

[TABLE]

for some real function $\phi$ with

[TABLE]

where $F\in C^{\infty}(M;\mathbb{R})$ satisfies that

[TABLE]

We call Equation (1.1) Calabi-Yau equation. Yau solved this equation by virtue of the continuity method. Consider the system of equations obtained by replacing $F$ by $tF+c_{t}$, where $c_{t}$ is a constant for $t\in[0,1]$ ($c_{0}=c_{1}=0$). An openness argument is obtained by the implicit function theorem and a closeness is derived by a priori estimate. Similar questions are proposed in symplectic manifolds in different cases and studied by many authors [4, 5, 12, 14].

Suppose that $(M,\omega)$ is a closed $2n$-dimensional symplectic manifold with a volume form $\sigma\in[\omega^{n}]$. In [11], Moser proved that there exists a symplectic form $\omega^{\prime}\in[\omega]$ which is symplectomorphic to $\omega$ satisfying that

[TABLE]

That is, there exists a diffeomorphism,

[TABLE]

isotopic to the identity such that

[TABLE]

It is easy to see that $\omega^{\prime}=f^{*}\omega$ is cohomologous to $\omega$ and $f^{*}\omega$ satisfies Equation (1.2). This is independent of the almost complex structures. It is well known that there are many almost complex structures which are compatible with the symplectic form $\omega$ and they form a contractible space [2, 10]. Here $\omega$ is compatible with an almost complex structure $J$, that is, at every point $p\in M$, $\omega_{p}(v,Jv)>0$ for every nonzero vector $v\in T_{p}M$ and $\omega(JY,JZ)=\omega(Y,Z)$ for all vector fields $Y$ and $Z$.

Now we suppose that $(M,g,J,\omega)$ is an almost Kähler manifold, that is, the symplectic form $\omega$ is compatible with the almost complex structure $J$ and $g(X,Y)=\omega(X,JY)$. Obviously, $g$ is a Riemannian metric. Consider the existence of the solution of Equation (1.2) in the following form,

[TABLE]

for

[TABLE]

Here

[TABLE]

and $\omega^{\prime}$ tames $J$, that is, at every point $p\in M$, $\omega^{\prime}_{p}(v,Jv)>0$ for every nonzero vector $v\in T_{p}M$. More precisely, this question can be expressed as follows:

Does there exist a smooth function,

[TABLE]

satisfying the following conditions?

[TABLE]

where $\sigma$ is a given volume form in $[\omega^{n}]$. Following a suggestion of M. Gromov, P. Delanoë studied this problem in [4].

We call Equation (1.4) generalized Calabi-Yau equation. In particular, if $(M,g,J,\omega)$ is Kähler (that is, $J$ is integrable), then,

[TABLE]

This implies that the form $\omega^{\prime}$ is a Kähler form. So Equation (1.4) reduces to the Calabi-Yau equation on Kähler manifolds, which was solved by S. T. Yau [16].

We define an operator $F$ from $C^{\infty}(M;\mathbb{R})$ to $C^{\infty}(M;\mathbb{R})$ as follows:

[TABLE]

where

[TABLE]

Therefore, Equation (1.4) is equivalent to the following problem:

Suppose that a positive function $f\in C^{\infty}(M;\mathbb{R})$ satisfies the following equality,

[TABLE]

Does there exist a solution of $\phi\in C^{\infty}(M;\mathbb{R})$ which satisfies the following equation?

[TABLE]

We need some notations in [4]:

Definition 1.1**.**

Suppose that $(M,g,J,\omega)$ is an almost Kähler manifold of real dimension $2n$. The sets $A$, $B$, $A_{+}$ and $B_{+}$ are defined as follows:

[TABLE]

Note that $A_{+}$ can be regarded as a convex open set of symplectic potential functions (analogue of Kähler potential functions). Restricting the operator $F$ to $A_{+}$, we get

[TABLE]

Thus, the existence of a solution to Equation (1.6) is equivalent to that the restricted operator

[TABLE]

is surjective.

Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of real dimension $2n$. If $J$ is integrable, then $F:A_{+}\rightarrow B_{+}$ is surjective. Conversely, if $F:A_{+}\rightarrow B_{+}$ is bijective and $n=2$, then $J$ is integrable. Delanoë [4] proved this result by constructing a suitable smooth function $\phi_{0}$ on the boundary of $A_{+}$ and he conjectured [4, Conjecture p.837] that the same result holds when $n\geq 2$.

In this paper, we prove Delanoë’s conjecture. Since an oriented surface is Kähler, we always consider the case $n\geq 2$.

Theorem 1.2**.**

Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of real dimension $2n$. Then the restricted operator $F|_{A_{+}}:A_{+}\rightarrow F(A_{+})$ is a diffeomorphism. Moreover, the restricted operator $F|_{A_{+}}:A_{+}\rightarrow B_{+}$ is a surjectivity map if and only if $J$ is integrable.

Remark 1.3**.**

1)* S. K. Donaldson gave a conjecture in [5]. Suppose that $J$ is an almost complex structure on a closed symplectic manifold $(M,\omega)$ of dimension $4$, and is tamed by the symplectic form $\omega$. Let $\sigma$ be a smooth volume form on $M$ with*

[TABLE]

Let $\tilde{\omega}$ be an almost Kähler form corresponding to the almost complex structure $J$ (that is, $\tilde{\omega}$ is a symplectic form compatible with $J$) with $[\tilde{\omega}]=[\omega]$, and solve the Calabi-Yau equation

[TABLE]

*Donaldson conjectured that there are $C^{\infty}$ a prior bounds on $\tilde{\omega}$ depending only on $\omega$, $J$ and $\sigma$. This is related to his broader program [5].

2)* V. Tosatti, B. Weinkove and S. Y. Yau [12] proved that Donaldson’s conjecture on estimates for Calabi-Yau equation in terms of a taming symplectic form can be reduced to an integral estimate of a scalar potential function. We will study Donaldson’s conjecture in a future paper [13].*

This article is organized as follows. Section 2 contains a local theory of the operator $F$ (Equality (1.5)) and the local expression of $\omega(\phi)$ (Equality (1.3)) by choosing the second canonical connection. Section 3 gives the proof of Theorem 1.2. In the following, we simply call Equation (1.4) the Calabi-Yau equation.

2 Local theory of Calabi-Yau equation

This section is devoted to a local theory of Calabi-Yau equation. Let $(M,g,J,\omega)$ be a closed almost Kähler manifold of real dimension $2n$. The $\mathbb{C}$-extension of the almost complex structure $J$ on the complexified tangent bundle $TM\otimes\mathbb{C}$ is defined by

[TABLE]

for $X,Y\in TM.$ Obviously,

[TABLE]

on $TM\otimes\mathbb{C}$. Therefore,

[TABLE]

where

[TABLE]

$J$ induces an almost complex structure on $\wedge^{k}T^{*}M\otimes\mathbb{C}$. Hence

[TABLE]

where $T_{p,q}$ is the space of $(p,q)$-forms. Let $P_{2,0}$, $P_{1,1}$ and $P_{0,2}$ be the projections to $T_{2,0}$, $T_{1,1}$ and $T_{0,2}$, respectively. For each $\phi\in C^{\infty}(M;\mathbb{R})$, set

[TABLE]

[TABLE]

Proposition 2.1**.**

Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of real dimension $2n$. Then $\omega(\phi)$ tames $J$ if and only if $H(\phi)$ tames $J$.

Proof.

From the definition of $\omega(\phi)$ and $H(\phi)$, we have

[TABLE]

The last equality holds since $\tau(\phi)(X,JX)=0$, for every vector field $X\in TM$. ∎

Remark 2.2**.**

Observe that the taming $(1,1)$-form $H(\phi)$ in Proposition 2.1 is not necessarily closed; it is, when $J$ is integrable (Kähler case) in which case it coincides with $\omega(\phi)$.

To compute $\tau(\phi)$ and $H(\phi)$, we choose a local coordinate system and the second canonical connection on an almost Hermitian manifold, $(M,g,J,\omega)$ of real dimension $2n$, (that is, a Riemannian metric $g$ is $J$-invariant and $\omega(X,Y)=g(JX,Y)$. ).

There exists a local orthonormal basis $\{\epsilon_{1},J\epsilon_{1},\cdots,\epsilon_{n},J\epsilon_{n}\}$ of $M$. Set $e_{j}=\frac{\epsilon_{j}-\sqrt{-1}J\epsilon_{j}}{\sqrt{2}}$. Then $\{e_{1},\cdots,e_{n},\bar{e}_{1},\cdots,\bar{e}_{n}\}$ is a local basis of $TM\otimes\mathbb{C}$ and $g(e_{i},e_{j})=g(\bar{e}_{i},\bar{e}_{j})=0,$ $g(e_{i},\bar{e}_{j})=\delta_{ij}.$ Obviously, $\{e_{1},\cdots,e_{n}\}$ is a local basis of $T^{1,0}.$ Let $\{\theta^{1},\cdots,\theta^{n}\}$ be its dual basis. Obviously, after complexification, $J$, $g$, and $\omega$ can be expressed locally as follows,

[TABLE]

[TABLE]

[TABLE]

Now, although this choice is not essential for the proof, we choose a different connection than the one used in [4], namely we choose the second canonical connection $\nabla^{1}$ (an almost Hermitian connection) on an almost Kähler manifold [6, 7, 12, 17]. Note that the second canonical connection on the almost Hermitian manifold $(M,g,J,\omega)$ is an affine connection $\nabla^{1}$ satisfying

[TABLE]

and the $(1,1)$-component of its torsion vanishes [7, 12, 17]. We must redo with $\nabla^{1}$ calculations analogous to those presented in [4, Appendix 1]. From Equality (2.4), we obtain that

[TABLE]

The second canonical connection, $\nabla^{1}$, induces

[TABLE]

Let $\phi\in C^{\infty}(M;\mathbb{R})$. Then

[TABLE]

Thus,

[TABLE]

Let

[TABLE]

Then, equality $d^{2}\phi=0$ and Equation (2.5) imply that

[TABLE]

Let

[TABLE]

be the torsion of the second canonical connection $\nabla^{1}$. Thus, $\Theta^{\alpha}$ contains $(2,0)$ and $(0,2)$ components only. Therefore,

[TABLE]

with $T_{\beta\gamma}^{\alpha}=-T_{\gamma\beta}^{\alpha}$ and $N_{\bar{\beta}\bar{\gamma}}^{\alpha}=-N_{\bar{\gamma}\bar{\beta}}^{\alpha}$. Indeed, the $(0,2)$ component of the torsion is independent of the choice of a metric and can be regarded as the Nijenhuis tensor, $N,$ of the almost complex structure $J$ [12, 17]. From Equations (2.3) and (2.9), we obtain that

[TABLE]

Suppose that $(M,g,J,\omega)$ is an almost Kähler manifold. By Equation (2), we have

[TABLE]

and

[TABLE]

Combining with Equations (2.6)$-$(2.10), we obtain that

[TABLE]

The reality of $dJd\phi$ implies that

[TABLE]

and

[TABLE]

Set

[TABLE]

Then,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Where $\delta_{\alpha\bar{\beta}}=1$ if $\alpha=\beta$; otherwise, $\delta_{\alpha\bar{\beta}}=0$. Hence, we have the following lemma:

Lemma 2.3**.**

Suppose that $(M,g,J,\omega)$ is a $2n$-dimensional almost Kähler manifold, and $\phi$ is a smooth real function on $M$. Then,

[TABLE]

where $\tau(\phi)$ and $H(\phi)$ are given by Equations (2.13) and (2.14), respectively.

Let $[\frac{n}{2}]$ denote the integral part of a positive number $\frac{n}{2}$. Following [4], let us define operators $F_{j}$ ($j=0,1,\cdots,[\frac{n}{2}]$) as follows:

[TABLE]

where

[TABLE]

Then

[TABLE]

The following result was proved in [4, Proposition 5]:

Proposition 2.4**.**

Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of real dimension $2n$. Then

[TABLE]

*If $\phi\in A_{+},$ then $F_{0}(\phi)>0$ and $F_{j}(\phi)\geq 0$, for $j=1,2,\cdots,[\frac{n}{2}]$.

For any $\phi\in A_{+}$, it is easy to see that the tangent space at $\phi$, $T_{\phi}A_{+}$, is $A$. For $u\in T_{\phi}A_{+}=A$, define $L(\phi)(u)$ by,

[TABLE]

For a local theory of Calabi-Yau equation, we should study the tangent map of the restricted operator $F|_{A_{+}}$. The following result was obtained in [4, Proposition 1]:

Lemma 2.5**.**

Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of dimension $2n$. Then the restricted operator

[TABLE]

is elliptic type on $A_{+}$. Moreover, the tangent map,

[TABLE]

of $F$ at $\phi\in A_{+}$ is a linear elliptic differential operator of second order without zero-th term.

The following result was proved in [4, Theorem 2]:

Proposition 2.6**.**

Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of dimension $2n$. Then the restricted operator

[TABLE]

is a diffeomorphic map.

Remark 2.7**.**

From the definition of $F(\phi)$ (Equality (1.5)), we see that

[TABLE]

Thus, $F(0)=1$. Therefore, Proposition 2.6 implies that the solution to Equation (1.4) exists and is unique if $\sigma$ is a small perturbation of $\omega^{n}$.

3 Global theory of Calabi-Yau equation

In this section, we give a proof of the main theorem. Let $(M,g,J,\omega)$ be a closed almost Kähler manifold of real dimension $2n$. If the almost complex structure $J$ is integrable, then the surjectivity of the restricted operator (1.7) is equivalent to the existence of solutions of Calabi-Yau equation on Kähler manifolds which was solved by S. T. Yau [16]. Suppose that $J$ is not integrable. We will prove that the restricted operator (1.7) is not surjective. For the case $n=2$, it was proved by P. Delanoë [4]. Now we give the proof for the case $n\geq 2$. More precisely, as in [4, p.835], we want to construct a function $\phi_{0}$ belonging to the boundary of the convex open set $A_{+}$ such that $F(\phi_{0})\in B_{+}$. First, let us give the definition of a pseudo holomorphic function [1, 2, 9].

Definition 3.1**.**

A smooth complex-valued function $f$ on a manifold $M$ is a pseudo holomorphic function at some point $p\in M$ if $df\circ J=idf$ at $p$.

To construct $\phi_{0}$ belonging to the boundary of $A_{+}$ and satisfying inequality $F(\phi_{0})>0$, we need the following lemmas:

Lemma 3.2**.**

Let $(M,J)$ be an almost complex manifold. Suppose that $f$ is a pseudo holomorphic function at some point $p\in M$. Then, for all vector fields $X,Y$,

[TABLE]

at $p$, where $N$ is the Nijenhuis tensor.

Proof.

If $X,Y\in T^{1,0}$, then

[TABLE]

Thus

[TABLE]

If

[TABLE]

then

[TABLE]

Combining with Equality (3), we get

[TABLE]

at $p\in M$. If $X,Y\in T^{0,1}$, then

[TABLE]

at $p\in M$. If $X\in T^{0,1}$ and $Y\in T^{1,0}$, then $N(X,Y)=0.$ Hence,

[TABLE]

at $p\in M$. The proof is completed. ∎

Remark 3.3**.**

Note that if there exist $n$ pseudo holomorphic functions on a real $2n$-dimensional almost Hermitian manifold $(M,g,J)$ which are independent at some point $p\in M$, then the Nijenhuis tensor $N$ identically vanishes at $p$. This means that an integrable complex structure is one with many (pseudo)-holomorphic functions. It is a hard theorem (Newlander-Nirenberg integrability theorem for almost complex structures) that the converse is also true. In general, an almost complex manifold has no holomorphic functions at all. On the other hand, it has a lot of pseudo-holomorphic curves (i.e., maps $f:\mathbb{C}\rightarrow(M^{2n},g,J)$ such that $df\circ i=J\circ df$) [9].

Let $(M,g,J,\omega)$ be a closed almost Kähler manifold of real dimension $2n$, where the almost complex structure $J$ is not integrable. Hence there exists a neighborhood $U_{0}$ of some point $p_{0}\in M$ where the Nijenhuis tensor $N$ does not vanish. Pick $f=\phi_{1}+i\phi_{2}$, with $\phi_{1}$ and $\phi_{2}$ real-valued, a complex function on $M$ which is not holomorphic on $U_{0}$. By Lemma 3.2, it satisfies

[TABLE]

on the neighborhood $U_{0}$. Without loss generality, we can suppose that for some $1\leq\alpha<\beta\leq n$,

[TABLE]

on the neighborhood $U_{0}$ of $p_{0}$. By Equality (2.15), there exists a suitable (non-zero) constant $c$ such that $h(c\phi_{1})$, defined in Section 2 (Equalities (2.12) and (2.15)), is a positive definite Hermitian matrix on $M$. Set $\phi=c\phi_{1}$. Then $\tau(\phi)$, defined in Section 2 (Equalities (2.1) and (2.13)) is non-zero on the neighborhood $U_{0}$ of $p_{0}$. So we obtain the following lemma:

Lemma 3.4**.**

*Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of real dimension $2n$, where the almost complex structure $J$ is not integrable. There exists a function $\phi\in C^{\infty}(M;\mathbb{R})$ satisfying the following conditions:

(1) $h(\phi)$ is a positive definite Hermitian matrix on $M$;

(2) there is a neighborhood $U_{0}$ of a point $p_{0}\in M$ such that $\tau(\phi)$ is non-zero on the neighborhood $U_{0}$.*

We can suppose that

[TABLE]

at $p_{0}$ [15]. By Lemma 3.4, we have that $\lambda_{\alpha}(0)>0$, for $\alpha=1,\cdots,n$. There exists a local coordinate system $\varphi^{-1}:U_{0}\rightarrow\mathbb{C}^{n}$ such that $\varphi(0)=p_{0}$ and

[TABLE]

at $p_{0}$, for $\alpha=1,\cdots,n$. Without loss generality, we can suppose that there is a polydisk $\triangle$ with center $p_{0}$:

[TABLE]

and there is $\epsilon_{1}>0$ such that, on $\varphi(\triangle)$,

[TABLE]

Let $\tilde{\eta}(r)$ be a $C^{2}$ cut-off function such that

[TABLE]

Given a real number $R>2$, let

[TABLE]

and

[TABLE]

Define a function $\psi_{R}$ as follows,

[TABLE]

where $z\in\triangle$. In the sequel, for short, we will abusively denote by $z_{i}$, $\eta_{R}$, $\Phi_{R}$, $\psi_{R}$, $\sigma$, the corresponding functions on $U_{0}\subset M$ obtained by composition with $\varphi^{-1}$. With this abuse of notation, recalling (2.7), we obtain by a direct calculation that

[TABLE]

with

[TABLE]

where $\Gamma_{\alpha\bar{\beta}}^{\gamma}=\omega_{\alpha}^{\gamma}({\bar{e}}_{\beta})$ are connection coefficients of the second canonical connection bounded on $\triangle$. Set

[TABLE]

Obviously, $\triangle_{R}\subset\triangle$. Henceforth, we will freely denote by the same letter $C$ various positive constants independent of $R>2$.

Lemma 3.5**.**

*Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of real dimension $2n$, and $\tilde{\eta}$, $\psi_{R}$, $\triangle$ and $U_{0}$ are defined as before. There exists a constant $C$ (depending only on $\lambda_{1}(0)$, $\lambda_{2}(0)$, $\tilde{\eta}$, $\omega$, $g$ and $J$) such that,

(1) on $\triangle$, for $1\leq\alpha\leq n$,*

[TABLE]

(2)* on $\triangle$, for $1\leq\alpha=\beta\leq 2$,*

[TABLE]

otherwise,

[TABLE]

where $\sigma$ is a continuous function (independent of $R$) and $\sigma(0)=0$. Moreover, at the center of $\triangle$, $|(\psi_{R})_{\alpha\bar{\beta}}|=\frac{\lambda_{\alpha}(0)}{2}$, for $1\leq\alpha=\beta\leq 2$; otherwise, $|(\psi_{R})_{\alpha\bar{\beta}}|=0$.

Proof.

From (3.5), we can see that $\psi_{R}$ is zero on $\triangle\setminus\triangle_{R}$. Hence it suffices to prove that the desired estimate holds on $\triangle_{R}$. Note that, for $1\leq\alpha,\beta,i\leq n$, the functions $|e_{\alpha}z_{i}|$, $|e_{\alpha}{\bar{z}}_{i}|$, $|{\bar{e}}_{\beta}z_{i}|$, $|{\bar{e}}_{\beta}{\bar{z}}_{i}|$, $|{\bar{e}}_{\beta}e_{\alpha}z_{i}|$ and $|{\bar{e}}_{\beta}e_{\alpha}{\bar{z}}_{i}|$ are continuous and bounded on $\Delta$. From (3.6) and (3.7), we have on $\triangle_{R}$

[TABLE]

A direct computation shows that

[TABLE]

where the hat means that the term is omitted. It is easy to see that

[TABLE]

Since

[TABLE]

we obtain that

[TABLE]

Thus

[TABLE]

Now we consider

[TABLE]

A direct computation shows that

[TABLE]

Thus,

[TABLE]

A direct computation shows that

[TABLE]

If $\alpha=\beta=1$ or $\alpha=\beta=2$, then

[TABLE]

If $3\leq\alpha=\beta\leq n$ or $1\leq\alpha\neq\beta\leq n$, then

[TABLE]

where $\sigma$ is the continuous function on $\triangle$ such that $\sigma(0)=0$, defined by:

[TABLE]

From (3.11)-(3.19), we obtain that

[TABLE]

where $\sigma$ is a continuous function and $\sigma(0)=0$. Besides, by (3.3) combined with (3.5) and (3), we obtain at $z=0$ that $|(\psi_{R})_{\alpha\bar{\alpha}}|(0)=\frac{\lambda_{\alpha}(0)}{2}$, for $1\leq\alpha\leq 2$; otherwise, $|(\psi_{R})_{\alpha\bar{\beta}}|(0)=0$. ∎

Lemma 3.6**.**

Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of real dimension $2n$, where the almost complex structure $J$ is not integrable. Suppose $\phi$, $\psi_{R}$, $\varphi$, $\tilde{\eta}$, $\triangle$ and $U_{0}$ are defined as before. Let $s\in[0,1]$. Then there exists $R_{1}>2$ independent of $s\in[0,1]$ such that, for each $R\geq R_{1}$, on $\varphi(\triangle_{R})$,

[TABLE]

where $\epsilon_{1}>0$ is the constant occuring in (3.4).

Proof.

Combining Lemma 3.5 with

[TABLE]

on $\varphi(\triangle_{R})$, we have the following inequality,

[TABLE]

By Lemma 3.4 and Inequality (3.4), we obtain,

[TABLE]

We can choose $R_{1}>2$ such that $\epsilon_{1}-\frac{C}{R_{1}}\geq\frac{\epsilon_{1}}{2}$. Thus, for each $R\geq R_{1}$, the desired estimate follows. ∎

For $s\in[0,1]$, we consider the function $\phi+s\psi_{R}\circ\varphi^{-1}$. If $s=0$, then

[TABLE]

is a positive definite Hermitian metric on $M$. As observed above, we have

[TABLE]

which, recalling (3.2), yields:

[TABLE]

The latter is a semi-positive definite Hermitian matrix at the center of $\triangle_{R}$ (i.e., $p_{0}=\varphi(0)\in M$). Following [4, Definition 3], let us recall the notion of positivity amplitude:

Definition 3.7**.**

Suppose that $(M,g,J,\omega)$ is a closed almost Hermitian manifold of real dimension $2n$. For each non-constant function $\phi\in C^{\infty}(M;\mathbb{R})$, the positivity amplitude of $\phi$ is the real number

[TABLE]

where $h(\phi)$ is defined in Section 2 (Equalities (2.12) and (2.15)).

Since $M$ is closed and the support of $\psi_{R}$ is compact, the positivity amplitude is finite. Hence, by Lemmas 3.4$-$3.6 and Equalities (3.20), (3.22), we obtain the following proposition:

Proposition 3.8**.**

Suppose that $(M,g,J,\omega)$ is a closed almost Kähler manifold of real dimension $2n$, where $J$ is not integrable. Suppose $\triangle_{R}$, $\psi_{R}$, $R_{1}$, $\phi$, $\varphi$, $U_{0}$ ,$\triangle$ and $a_{\omega}(\phi)$ are defined as before and set

[TABLE]

*Then, for each $R\geq R_{1}$, we have:

(1) $0<a_{\omega(\phi)}(\psi_{R}\circ\varphi^{-1})\leq 1$;

(2) $\phi_{R}$ lies on the boundary of $A_{+}$;

(3) $h(\phi+a_{\omega(\phi)}(\psi_{R}\circ\varphi^{-1})\cdot\psi_{R}\circ\varphi^{-1})$ is a semi-positive definite Hermitian matrix function on $\varphi(\triangle_{R})$ and a positive definite Hermitian matrix function on $M\setminus\varphi(\triangle_{R})$.*

Now we will prove the following key lemma, sticking to the assumptions of Proposition 3.8:

Lemma 3.9**.**

There exists $R_{0}\geq R_{1}$ large enough such that the function $\phi_{0}=\phi_{R_{0}}$ satisfies $F(\phi_{0})>0$ on $M$.

Proof.

Note that

[TABLE]

From (2.16), we have at $p_{0}$

[TABLE]

From Equalities (3.2), (3.21), Lemma 3.6 and Proposition 3.8, we have, at $p_{0}$, that

[TABLE]

Let

[TABLE]

where

[TABLE]

and

[TABLE]

Note that $H(\phi)_{\alpha\bar{\alpha}}(p_{0})=\lambda_{\alpha}(0)$, for $1\leq\alpha\leq n$. $H(\phi)_{\alpha\bar{\beta}}(p_{0})=0$, for $\alpha\neq\beta$. Let us define a neighborhood of $0\in\triangle$ by:

[TABLE]

for some $\delta\in(0,\frac{1}{2})$. Recalling (2.16), we have

[TABLE]

By Lemma 3.5, we obtain that, on $\triangle_{R}\bigcap V_{\delta}$, each component of $H^{\prime}$ is less than $C(\delta+\frac{1}{R})$ and $|(\tau(\phi_{R}))_{\alpha\beta}|\leq C$, where $C$ is independent of $R$. By Lemma 3.4, we have that, on $\triangle_{R}\cap V_{\delta}$,

[TABLE]

where $C$ is independent of $R$ and $\alpha=3,\cdots,n$. By Lemma 3.5 and Proposition 3.8, we have that, on $\triangle_{R}\cap V_{\delta}$,

[TABLE]

where $\alpha=1,2$ and $C$ is independent of $R$. Therefore, on $\triangle_{R}\cap V_{\delta}$, recalling the positivity lemma of [4, Appendix 3], we have

[TABLE]

where $C$ and $\epsilon_{0}$ is independent of $R$. Choose

[TABLE]

Thus, for each $R>R_{2}$,

[TABLE]

on $V_{\delta_{1}}\bigcap\triangle_{R}$. Moreover, we can choose $R_{0}>R_{2}$ such that $\triangle_{R_{0}}\subset V_{\delta_{1}}$. Define

[TABLE]

By construction, we have $\phi_{0}=\phi$ on $M\setminus\varphi(\triangle_{R_{0}})$ and, on $\varphi(\triangle_{R_{0}})$:

[TABLE]

and also [4, Appendix 3]: $\forall j\geq 2,F_{j}(\phi_{0})\geq 0$. Therefore, by Proposition 2.4, we conclude that $F(\phi_{0})>0$ on $M$, or else $F(\phi_{0})\in B_{+}$. ∎

Proof of Theorem 1.2    For completeness, let us redo the argument of [4]. First, Proposition 2.6 has shown that $F|_{A_{+}}:A_{+}\rightarrow F(A_{+})$ is a diffeomorphism. If $J$ is integrable, then $(M,g,J,\omega)$ is a closed Kähler manifold. The restricted operator $F|_{A_{+}}$ is a surjectivity map since there always exists a solution of Calabi-Yau equation on Kähler manifold [16]. Suppose that $J$ is not integrable. From Lemma 3.9, we have constructed a function $\phi_{0}$ belonging to the boundary of $A_{+}$ and $F(\phi_{0})>0$ on $M$. We claim that $F(\phi_{0})$ does not belong to the image of $F$ on $A_{+}$. Otherwise, we may assume that there exists a function $\phi_{1}\in A_{+}$ such that

[TABLE]

For $t\in(0,1]$, let

[TABLE]

Obviously, for all $t\in(0,1],$ $\phi_{t}\in A_{+}$. Hence

[TABLE]

But

[TABLE]

Then, by Lemma 2.5, we have that

[TABLE]

is a linear elliptic operator of second order without zero-th term. Making use of maximal principle of Hopf, we obtain that the kernel of $L(\phi_{0})$ consists of constant functions. Therefore,

[TABLE]

(note that $\phi_{0}$, $\phi_{1}$ belong to $\bar{A}_{+}$) which implies that

[TABLE]

This contradicts the fact that $\phi_{0}$ belongs to the boundary of $A_{+}$.

Acknowledgements The authors would like to thank Xifang CAO, Jinquan LUO and Ying ZHANG for helpful discussions. The authors also express their sincere gratitude toward the referee for having pointed out several nice and pertinent improvements in the writing of the paper.