The Monge-Ampère equation for $(n-1)$ -quaternionic PSH functions on a hyperKähler manifold

Jixiang Fu and Xin Xu and Dekai Zhang Shanghai Center for Mathematical Sciences, Jiangwan Campus, Fudan University, Shanghai, 200438, China majxfu@fudan.edu.cn School of Mathematical Sciences, Fudan University, Shanghai 200433, China 20110180015@fudan.edu.cn Department of Mathematics, Shanghai University, Shanghai, 200444, China dkzhang@shu.edu.cn

Abstract.

We prove the existence of unique smooth solutions to the quaternionic Monge-Ampère equation for $(n-1)$ -quaternionic plurisubharmonic functions on a hyperKähler manifold and thus obtain solutions for the quaternionic form type equation. We derive the $C^{0}$ estimate by establishing a Cherrier-type inequality as in Tosatti and Weinkove [22]. By adopting the approach of Dinew and Sroka [9] to our context, we obtain $C^{1}$ and $C^{2}$ estimates without assuming the flatness of underlying hyperKähler metric comparing to previous results [14].

1. Introduction

A hypercomplex manifold is a smooth manifold $M$ together with a triple $(I,J,K)$ of complex structures satisfying the quaternoinic relation

IJ=-JI=K.

Let $(M,I,J,K)$ be a hypercomplex manifold, and $g$ a Riemannian metric on $M$ . The metric $g$ is called hyperhermitian if $g$ is hermitian with respect to $I,J,K$ , i.e. for any vector fields $X$ and $Y$ on $M$ ,

g(X,Y)=g(XI,YI)=g(XJ,YJ)=g(XK,YK).

Denote by $\omega_{I},\omega_{J},\omega_{K}$ the fundamental form corresponding to $I,J,K$ respectively and let $\Omega=\omega_{J}+i\omega_{K}$ . On a hyperhermitian manifold $(M,I,J,K,g)$ , the metric $g$ is called hyperKähler (HK) if $d\Omega=0$ or equivalently $d\omega_{I}=d\omega_{J}=d\omega_{K}=0$ , and called hyperKähler with torsion (HKT) if $\partial\Omega=0$ .

In analogy with the classical Calabi-Yau theorem [27] on the complex Monge-Ampère equation on a Kähler manifold, Alesker and Verbitsky [4] conjectured the existsnce of solutions to the quaternoinic Monge-Ampère equation on a compact HKT manifold of quaternionic dimension $n$ . It takes the form

(1.1)

\begin{split}(\Omega+\partial\partial_{J}u)^{n}&=e^{f}\Omega^{n},\\ \Omega+\partial\partial_{J}u&>0,\end{split}

where $\partial_{J}=J^{-1}\circ\overline{\partial}\circ J$ . While general solution to this equation remains open, partial results can be found in [1, 5, 13, 9, 4, 2, 3, 18]. Specifically, Alesker and Verbitsky [4] obtained $C^{0}$ estimate when the canonical bundle is holomorphically trivial. Alesker [1] proved the conjecture on compact manifolds with a flat hyperKähler metric. In [2] Alesker and Shelukhin proved $C^{0}$ estimate without any extra assumptions and the proof was later simplyfied by Sroka [18]. Dinew and Sroka [9] solved equation (1.1) on a hyperKähler manifold.

As in the complex setting, we consider the quaternoinic form-type equation, as the analogue of the form-type equation which was proposed and also solved on a Kähler manifold of nonnegative bisectional curvature by Fu-Wang-Wu [10, 11]. It was later shown by Tosatti and Weinkove [22] that the assumption on curvature can be removed.

In particular, one can define quaternionic balanced metrics on hypercomplex manifolds by $\partial\Omega^{n-1}=0$ (see [15]). Let $(M,I,J,K,g,\Omega)$ be a hypercomplex manifold of quaternionic dimension $n$ , and $g_{0}$ a quaternionic balanced metric on $M$ with induced $(2,0)$ -form $\Omega_{0}$ . Let $\varphi$ be a $(2n-4,0)$ -form such that $\Omega_{0}^{n-1}+\partial\partial_{J}\varphi$ is strictly positive. Then there exists a quaternionic balanced metric $\Omega_{\varphi}$ such that

(1.2)

\Omega_{\varphi}^{n-1}=\Omega_{0}^{n-1}+\partial\partial_{J}\varphi.

The quaternionic form-type Calabi-Yau equation is written as

(1.3)

\Omega_{\varphi}^{n}=e^{f^{\prime}+b^{\prime}}\Omega^{n}

where $f^{\prime}$ is a given smooth function on $M$ and $b^{\prime}$ is a uniquely determined constant. Solving equation (1.3) gives a quaternionic balanced metric $\Omega_{\varphi}$ with prescribed volume form up to scaling. One can reduce the form-type equation to function type by considering a function $u\in C^{\infty}(M,\mathbb{R})$ such that $\Omega_{0}^{n-1}+\partial\partial_{J}(u\Omega^{n-2})$ is strictly positive, and denote by $\Omega_{u}$ the unique strictly positive $(2,0)$ -form such that

(1.4)

\Omega_{u}^{n-1}=\Omega_{0}^{n-1}+\partial\partial_{J}(u\Omega^{n-2}).

Then equation (1.3) is reduced to

(1.5)

\Omega_{u}^{n}=e^{f^{\prime}+b^{\prime}}\Omega^{n}.

In particular when $\Omega$ is HKT, $\partial\Omega=\partial_{J}\Omega=0$ . Then (1.4) becomes

(1.6)

\Omega_{u}^{n-1}=\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2}.

In this paper, we consider equation (1.5) under the assumption that $\Omega$ is hyperKähler. Parallel to the complex case in [22], equation (1.5) can be restated as equation (1.7) in terms of quaternoinic Monge-Ampère equation for $(n-1)$ -quaternoinic plurisubharmonic functions. Our main result is as follows.

Theorem 1.1.

Let $(M,I,J,K,g,\Omega)$ be a compact hyperKähler manifold of quaternionic dimension $n$ , and $\Omega_{h}$ a strictly positive $(2,0)$ -form with respect to $I$ . Let $f$ be a smooth function on $M$ . Then there is a unique pair $(u,b)\in C^{\infty}(M,\mathbb{R})\times\mathbb{R}$ , solving

(1.7)

\big{(}\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u)\big{)}^{n}=e^{f+b}\Omega^{n}

with

(1.8)

\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u)>0,\quad\sup_{M}u=0.

Here $S_{1}(\partial\partial_{J}u)$ is defined in Section 2 and related to the Chern Laplacian (see (2.4), (2.5)). Recently on a locally flat hyperhermitian manifold, Gentili and Zhang [14] studied a general class of fully non-linear equations including equation (1.7) and they solved the equation assuming the existence of a flat hyperKähler metric. Here we are able to remove the assumption on flatness. From Theorem 1.1 we obtain

Corollary 1.2.

Let $(M,I,J,K,g,\Omega)$ be a compact hyperKähler manifold of quaternionic dimension $n$ and $g_{0}$ a quaternionic balanced (resp., Gauduchon; resp., strongly Gauduchon) metric on $M$ with induced $(2,0)$ -form $\Omega_{0}$ . Then for a given smooth function $f^{\prime}$ on $M$ , there exists a unique constant $b^{\prime}$ and a unique quaternionic balanced (resp., Gauduchon; resp., strongly Gauduchon) metric $\Omega_{u}$ satisfying (1.6) and solving (1.5).

We obtain a priori estimates and thus employ the continuity method to prove Theorem 1.1. In Section 2 we give definitions and notations used throughout this paper, and explain the relation between equation (1.5) and (1.7). We derive $C^{0}$ estimate in Section 3 by establishing a Cherrier-type inequality without using the hyperKähler condition. We derive $C^{1}$ estimate in Section 4 and $C^{2}$ estimate in Section 5 and Section 6. Then the main theorem is proved in last section.

2. Preliminaries and notation

On a hypercomplex manifold $(M,I,J,K)$ , the exterior differential $d$ is decomposed into $d=\partial_{I}+\bar{\partial}_{I}$ with respect to complex structure $I$ . For simplicity we denote by $\partial=\partial_{I}$ and $\bar{\partial}=\bar{\partial}_{I}$ . Verbitsky [24] introduced the operator $\partial_{J}$ as the quaternionic analogue of $\bar{\partial}$ by

\partial_{J}=J^{-1}\circ\bar{\partial}\circ J.

As in [9] we also define

\bar{\partial}_{J}=J^{-1}\circ\partial\circ J.

The operators $\partial,\bar{\partial},\partial_{J}$ and $\bar{\partial}_{J}$ satisfy the following properties.

Lemma 2.1 ([9], Lemma 2.12).

For a hypercomplex manifold $(M,I,J,K)$ the following holds.

(2.1)

\begin{gathered}\partial^{2}=\bar{\partial}^{2}=\partial_{J}^{2}=\bar{\partial}_{J}^{2}=0,\\ \partial\bar{\partial}+\bar{\partial}\partial=\partial_{J}\bar{\partial}_{J}+\bar{\partial}_{J}\bar{\partial}=\partial\partial_{J}+\partial_{J}\partial=0,\\ \bar{\partial}\bar{\partial}_{J}+\bar{\partial}_{J}\bar{\partial}=\partial_{J}\bar{\partial}+\bar{\partial}\partial_{J}=\bar{\partial}_{J}\partial+\partial\bar{\partial}_{J}=0.\end{gathered}

On a hyperhermitian manifold $(M,I,J,K,g)$ of quaternionic dimension $n$ , let

\Omega=\omega_{J}+i\omega_{K}

where $\omega_{J}$ and $\omega_{K}$ are the fundamental forms of $(g,J)$ and $(g,K)$ respectively. We denote by $\bigwedge_{I}^{p,q}(M)$ the $(p,q)$ -forms with respect to $I$ , which we simply call $(p,q)$ -forms throughout this paper. A form $\alpha\in\bigwedge_{I}^{2k,0}(M)$ satisfying $J\alpha=\overline{\alpha}$ is called $J$ -real and denoted by $\alpha\in\bigwedge_{I,\mathbb{R}}^{2k,0}(M)$ . In particular, we have $\Omega\in\bigwedge_{I,\mathbb{R}}^{2,0}(M)$ .

Definition 2.2.

A $J$ -real $(2,0)$ -form $\alpha$ is said to be positive (resp. strictly positive) if $\alpha(X,\overline{X}J)\geq 0$ (resp. $\alpha(X,\overline{X}J)>0$ ), for any non-zero $(1,0)$ -vector $X$ .

In complex case, one can simultaneously diagonalize two hermitian matrices when one of them is positive definete. Similar result holds for $J$ -real $(2,0)$ -forms.

Lemma 2.3 ([18], Lemma 3).

Let $\alpha$ and $\beta$ be two $J$ -real $(2,0)$ -forms on a hyperhermitian manifold $M$ of quaternionic dimension $n$ , and $\alpha$ is strictly positive. Then for each $x\in M$ there exists a basis $e_{1},\overline{e_{1}}J,\dots,e_{n},\overline{e_{n}}J$ of $T^{1,0}_{I,x}(M)$ such that

\alpha(e_{i},e_{j})=\beta(e_{i},e_{j})=\alpha(e_{i},\overline{e_{j}}J)=\beta(e_{i},\overline{e_{j}}J)=0\text{\, for\, }i\neq j.

Analogous to positive definite $(n-1,n-1)$ -form in complex case [22], we define strictly positive $(2n-2,0)$ -form as follows.

Definition 2.4.

A $J$ -real $(2n-2,0)$ -form $\Phi$ is said to be strictly positive if $\Phi\wedge\alpha\geq 0$ , for any positive $(2,0)$ -form $\alpha$ , with equality if and only if $\alpha=0$ . We denote all strictly positive $J$ -real $(2n-2,0)$ -forms by $\bigwedge_{I,\mathbb{R}}^{2n-2,0}(M)_{>0}$ .

The notion of (strictly) positive forms on hypercomplex manifolds can be found in various literature [3, 26, 25], and we refer readers to [26] for thorough discussions. For complex case see for instance [8].

As in [14], we define the Hodge star-type operator $\ast:\bigwedge_{I}^{p,0}(M)\to\bigwedge_{I}^{2n-p,0}(M)$ by the relation

\alpha\wedge\ast\beta=\frac{1}{n!}\langle\alpha,\beta\rangle_{g}\Omega^{n},\quad\text{for \,}\alpha,\beta\in\bigwedge\nolimits_{I}^{p,0}(M).

Here, the pointwise inner product $\langle\,,\,\rangle_{g}$ is defined by

\langle\alpha,\beta\rangle_{g}=\frac{1}{p!}\sum g^{\lambda_{1}\overline{\mu}_{1}}\cdots g^{\lambda_{p}\overline{\mu}_{p}}\alpha_{\lambda_{1}\cdots\lambda_{p}}\overline{\beta_{\mu_{1}\cdots\mu_{p}}},\quad\text{for \,}\alpha,\beta\in\bigwedge\nolimits_{I}^{p,0}(M)

where any $(p,0)$ -form $\alpha$ is locally written as

\alpha=\frac{1}{p!}\sum\alpha_{\lambda_{1}\cdots\lambda_{p}}dz^{\lambda_{1}}\wedge\cdots\wedge dz^{\lambda_{p}}.

At a point $p\in M$ we can take $I$ -holomorphic coordinates $(z^{0},\cdots,z^{2n-1})$ such that $(g_{\lambda\overline{\mu}})$ is the identity at $p$ , then we have

\ast(dz^{2i}\wedge dz^{2i+1})=dz^{0}\wedge dz^{1}\wedge\cdots\wedge\widehat{dz^{2i}}\wedge\widehat{dz^{2i+1}}\wedge\cdots\wedge dz^{2n-2}\wedge dz^{2n-1}.

It is easy to show that the operator $\ast$ maps $\bigwedge_{I,\mathbb{R}}^{2,0}(M)_{>0}$ to $\bigwedge_{I,\mathbb{R}}^{2n-2,0}(M)_{>0}$ and vice versa.

Definition 2.5.

(1)

For a $J$ -real $(2,0)$ -form locally written as $\alpha=\sum_{i<j}\alpha_{ij}dz^{i}\wedge dz^{j}$ , define the Pfaffian of $\alpha$ locally by $\textnormal{Pf}(\alpha)dz^{0}\wedge\cdots\wedge dz^{2n-1}=\alpha^{n}$ .
(2)

The Pfaffian of a $J$ -real $(2n-2,0)$ -form $\Phi$ is defined by

$\textnormal{Pf}(\Phi)=\textnormal{Pf}(\frac{1}{(n-1)!}\ast\!\Phi).$

In particular, we have for any $\alpha\in\bigwedge_{I,\mathbb{R}}^{2,0}(M)$ ,

(2.2)

\text{Pf}(\alpha^{n-1})=\text{Pf}(\alpha)^{n-1}.

In fact, computing at a point and using Lemma 2.3 we can write $\alpha=\sum\lambda_{i}dz^{2i}\wedge dz^{2i+1}$ . Define $\Lambda=\lambda_{0}\cdots\lambda_{n-1}$ , $\Lambda_{i}=\lambda_{0}\cdots\hat{\lambda_{i}}\cdots\lambda_{n-1}$ . Then $\text{Pf}(\alpha)=\Lambda$ . On the other hand,

\alpha^{n-1}=(n-1)!\sum\Lambda_{i}dz^{0}\wedge dz^{1}\wedge\cdots\wedge\widehat{dz^{2i}}\wedge\widehat{dz^{2i+1}}\wedge\cdots\wedge dz^{2n-2}\wedge dz^{2n-1}.

By definition we have $\text{Pf}(\alpha^{n-1})=\Lambda^{n-1}$ . Hence (2.2) follows.

Also, observe that for any two $J$ -real $(2,0)$ -forms $\chi$ and $\eta$ , we have

(2.3)

\frac{\chi^{n}}{\eta^{n}}=\frac{\text{Pf}(\chi)}{\text{Pf}(\eta)}=\frac{\text{Pf}(\ast\chi)}{\text{Pf}(\ast\eta)}.

For conveninence in later computation, we introduce the following definition.

Definition 2.6.

For $\chi\in\bigwedge_{I,\mathbb{R}}^{2,0}(M)$ , define

(2.4)

S_{m}(\chi)=\frac{C_{n}^{m}\chi^{m}\wedge\Omega^{n-m}}{\Omega^{n}}\quad\text{for}\quad 0\leq m\leq n.

In particular for $u\in C^{\infty}(M,\mathbb{R})$ we have

(2.5)

S_{1}(\partial\partial_{J}u)=\frac{1}{2}\Delta_{I,g}u.

In fact, choose local coordinates such that $\Omega=\sum_{i=0}^{n-1}dz^{2i}\wedge dz^{2i+1}$ . Now $\omega_{I}$ takes the form

\omega_{I}=\frac{i}{2}\sum_{\alpha=0}^{2n-1}dz^{\alpha}\wedge d\overline{z}^{\alpha}.

Since

J(\partial\partial_{J}u)=-J(\partial_{J}\partial u)=-\bar{\partial}J\partial u=\bar{\partial}J^{-1}\partial u=\bar{\partial}\bar{\partial}_{J}u=\overline{\partial\partial_{J}u},

we see that $\partial\partial_{J}u$ is $J$ -real. Then compute

\begin{split}\partial\partial_{J}u&=\sum\partial(J^{-1}\bar{\partial}u)=\sum\partial(u_{\bar{j}}J^{-1}d\bar{z}^{j})\\ &=\sum u_{\bar{j}i}dz^{i}\wedge J^{-1}d\bar{z}^{j}+\sum u_{\bar{j}}\partial(J^{-1}d\bar{z}^{j})=\sum u_{\bar{j}i}dz^{i}\wedge J^{-1}d\bar{z}^{j}.\end{split}

The last equality above is derived from

0=(\bar{\partial}\bar{\partial}_{J}+\bar{\partial}_{J}\bar{\partial})(z^{i})=\bar{\partial}J^{-1}\partial J(z^{i})+J^{-1}\partial J\bar{\partial}z^{i}=J^{-1}\partial Jd\bar{z}^{i}.

Hence

S_{1}(\partial\partial_{J}u)=\frac{n\partial\partial_{J}u\wedge\Omega^{n-1}}{\Omega^{n}}=\sum_{\alpha=0}^{2n-1}u_{\alpha\overline{\alpha}}

and

\Delta_{I,g}u=\frac{2n\partial\overline{\partial}u\wedge\omega_{I}^{2n-1}}{\omega_{I}^{2n}}=2\sum_{\alpha=0}^{2n-1}u_{\alpha\overline{\alpha}}.

Thus equation (2.5) holds.

Now let $(M,I,J,K,g,\Omega)$ be a compact hyperKähler manifold and $\Omega_{h}$ a strictly positive $(2,0)$ -form with respect to $I$ . The quaternionic Monge–Ampère equation for $(n-1)$ -quaternionic plurisubharmonic functions is written as

(2.6)		$\displaystyle\big{(}\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u)\big{)}^{n}=e^{f+b}\Omega^{n}$
(2.7)		$\displaystyle\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u)>0,\quad\sup_{M}u=0.$

For a quaternionic balanced metric (resp., Gauduchon; resp., strongly Gauduchon, for definitions and their correspondence with the complex case see [15, Table 2]) with induced $(2,0)$ -form $\Omega_{0}$ , we define $\Omega_{h}$ by

(2.8)

(n-1)!\ast\Omega_{h}=\Omega_{0}^{n-1}.

We would like to show that a solution to equation (2.6) gives rise to a solution to the quaternionic form type equation. We also need

(2.9)

\frac{1}{(n-1)!}\ast(\partial\partial_{J}u\wedge\Omega^{n-2})=\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u)

which can be seen by computing in local coordinates. We refer readers to [14, p. 34] for details. By (2.3), (2.8) and (2.9), we have

\begin{split}&\frac{\big{(}\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u)\big{)}^{n}}{\Omega^{n}}\\ =&\frac{\text{Pf}\big{(}\ast(\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u))\big{)}}{\text{Pf}(\ast\Omega)}\\ =&\frac{\text{Pf}(\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2})}{\text{Pf}(\Omega^{n-1})}.\end{split}

Now observe that a strictly positive $(2n-2,0)$ -form $\Phi$ can be written as $\Phi=\phi^{n-1}$ , where $\phi$ is a strictly positive $(2,0)$ form. The proof here is almost identical to the arguments in [16, p. 279-280]. Since $\ast$ maps $\bigwedge_{I,\mathbb{R}}^{2,0}(M)_{>0}$ to $\bigwedge_{I,\mathbb{R}}^{2n-2,0}(M)_{>0}$ , we have

(2.10)

\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2}>0.

Thus there exist $\Omega_{u}$ such that

\Omega_{u}^{n-1}=\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2}.

Such $\Omega_{u}$ is quaternionic balanced (resp., Gauduchon; resp., strongly Gauduchon) and we have

e^{f+b}=\frac{\text{Pf}(\Omega_{u}^{n-1})}{\text{Pf}(\Omega^{n-1})}=\frac{\text{Pf}(\Omega_{u})^{n-1}}{\text{Pf}(\Omega)^{n-1}}=\left(\frac{\Omega_{u}^{n}}{\Omega^{n}}\right)^{n-1}.

It follows that a solution to equation (2.6) solves

(2.11)

\Omega_{u}^{n}=e^{f^{\prime}+b^{\prime}}\Omega^{n}.

This proves Corollary 1.2.

Remark 2.7.

On a hyperhermitian manifold $(M,I,J,K,g)$ , There exists a unique torsion free connection $\nabla^{O}$ called Obata connection [17] such that

\nabla^{O}I=\nabla^{O}J=\nabla^{O}K=0.

It is well known that the hyperKähler condition $d\Omega=0$ is equivalent to $\nabla^{O}=\nabla^{LC}$ where $\nabla^{LC}$ is the Levi-Civita connection. Using Obata connection it is shown in [9, Sect. 2.4] that one can choose around any point $x\in M$ local $I$ -holomorphic geodesic coordinates such that the Christoffel symbol of $\nabla^{O}$ and the first derivatives of $J$ vanish at $x$ . This property is crucial for $C^{2}$ estimate in Sect. 6.

From above remark, we have the following useful lemma.

Lemma 2.8.

On a hyperhermitian manifold $(M,I,J,K,g)$ , given a $J$ -real $(2,0)$ -form $\alpha$ , for any point $x\in M$ , one can choose around $x$ local $I$ -holomorphic coordinates such that the following relations hold at $x$ , for all $i,j=0,\dots,n-1$ .

(2.12)

\begin{split}&\alpha_{2i2j,p}=\overline{\alpha_{2i+12j+1,\bar{p}}}\,,\quad\alpha_{2i2j+1,p}=\overline{\alpha_{2j2i+1,\bar{p}}}\,,\\ &\alpha_{2i+12j,p}=\overline{\alpha_{2j+12i,\bar{p}}}\,,\quad\alpha_{2i+12j+1,p}=\overline{\alpha_{2i2j,\bar{p}}}\,,\end{split}

if $\alpha$ is locally written as

\alpha=\sum_{i<j}\alpha_{ij}dz^{i}\wedge dz^{j},\quad\alpha_{ij}=-\alpha_{ji},

and

\alpha_{ij,p}=\frac{\partial}{\partial z^{p}}\alpha_{ij},\quad\alpha_{ij,\bar{p}}=\frac{\partial}{\partial\bar{z}^{p}}\alpha_{ij}.

Proof.

Choose local $I$ -holomorphic coordinates around $x$ such that at $x$ , the first derivatives of $J$ vanish and

Jdz^{2i}=-d\bar{z}^{2i+1},\quad Jdz^{2i+1}=d\bar{z}^{2i}.

The $J$ action on $1$ -forms is given by

Jdz^{i}=J^{i}_{\bar{k}}d\bar{z}^{k}.

Hence

\begin{split}J\alpha&=\sum_{k<l}\alpha_{kl}Jdz^{k}\wedge Jdz^{l}=\sum_{i,j}\sum_{k<l}\alpha_{kl}J^{k}_{\bar{i}}J^{l}_{\bar{j}}d\bar{z}^{i}\wedge d\bar{z}^{j}\\ &=\sum_{i<j}\sum_{k<l}\alpha_{kl}(J^{k}_{\bar{i}}J^{l}_{\bar{j}}-J^{k}_{\bar{j}}J^{l}_{\bar{i}})d\bar{z}^{i}\wedge d\bar{z}^{j}.\end{split}

Since the derivatives of $J$ vanish at $x$ , taking $\partial$ and evaluating at $x$ gives

(2.13)

\partial J\alpha=\sum_{p}\sum_{i<j}\sum_{k<l}\alpha_{kl,p}(J^{k}_{\bar{i}}J^{l}_{\bar{j}}-J^{k}_{\bar{j}}J^{l}_{\bar{i}})dz^{p}\wedge d\bar{z}^{i}\wedge d\bar{z}^{j}.

On the other hand

(2.14)

\partial\bar{\alpha}=\sum_{p}\sum_{i<j}\overline{\alpha_{ij,}}{}_{p}dz^{p}\wedge d\bar{z}^{i}\wedge d\bar{z}^{j}.

Notice at the point $x$

J^{2i}_{\overline{2i+1}}=-1,\quad J^{2i+1}_{\overline{2i}}=1,

and all the other $J^{k}_{\bar{l}}$ vanish. Since $J\alpha=\bar{\alpha}$ , comparing components of (2.13) and (2.14) we get for example, when $2i+1<2j+1$ ,

\begin{split}\overline{\alpha_{2i+12j+1,}}{}_{p}&=\sum_{k<l}\alpha_{kl,p}(J^{k}_{\overline{2i+1}}J^{l}_{\overline{2j+1}}-J^{k}_{\overline{2j+1}}J^{l}_{\overline{2i+1}})\\ &=\alpha_{2i2j,p}J^{2i}_{\overline{2i+1}}J^{2j}_{\overline{2j+1}}=\alpha_{2i2j,p}\,.\end{split}

And when $2j+1<2i$ ,

\begin{split}\overline{\alpha_{2j+12i,}}{}_{p}&=\sum_{k<l}\alpha_{kl,p}(J^{k}_{\overline{2j+1}}J^{l}_{\overline{2i}}-J^{k}_{\overline{2i}}J^{l}_{\overline{2j+1}})\\ &=\alpha_{2j2i+1,p}J^{2j}_{\overline{2j+1}}J^{2i+1}_{\overline{2i}}=-\alpha_{2j2i+1,p}=\alpha_{2i+12j,p}\,.\end{split}

By taking all the other combinations of $i,j$ we obtain (2.12). ∎

Remark 2.9.

Combining Lemma 2.3 and Remark 2.7, on a hyperhermitian manifold $(M,I,J,K,g,\Omega)$ of quaternionic dimension $n$ , we can find local $I$ -holomorphic geodesic coordinates suth that $\Omega$ and another $J$ -real $(2,0)$ -form $\widetilde{\Omega}$ are simultaneously diagonalizable at a point $x\in M$ , i.e.

\Omega=\sum_{i=0}^{n-1}dz^{2i}\wedge dz^{2i+1},\quad\widetilde{\Omega}=\sum_{i=0}^{n-1}\widetilde{\Omega}_{2i2i+1}dz^{2i}\wedge dz^{2i+1},

and the Christoffel symbol of $\nabla^{O}$ and first derivatives of $J$ vanish at $x$ , i.e.

J^{l}_{\bar{k},i}=J^{\bar{l}}_{k,i}=J^{\bar{l}}_{k,\bar{i}}=J^{l}_{\bar{k},\bar{i}}=0.

We call such local coordinates the normal coordinates around $x$ .

3. $C^{0}$ Estimate

Recently Sroka [19] obtained a sharp $C^{0}$ estimate for a class of PDEs given by the operator dominating the quaternionic Monge-Ampère operator. Here we adopt a different approach for our purpose by establishing a Cherrier-type inequality and the lemmas in [22]. We obtain

Theorem 3.1.

Let $(M,I,J,K,g,\Omega)$ be a compact hyperhermitian manifold of quaternionic dimension $n$ , and $\Omega_{h}$ a strictly positive $(2,0)$ -form with respect to $I$ . Let $f$ be a smooth function on $M$ . If $u$ is a solution to equation (1.7) satisfying (1.8). Then there exists a constant $C$ depending only on the fixed data $(I,J,K,g,\Omega,\Omega_{h})$ and $f$ such that

\sup_{M}|u|\leq C.

Notice that by maximum principal the constant $b$ in equation (1.7) is uniformly bounded by $\sup_{M}|f|$ , $\Omega$ and $\Omega_{h}$ . In fact, at the maximum point of $u$ ,

S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u\leq 0.

Hence by equation (1.7) $b$ is bounded above. Similarly $b$ is also bounded below. Thus for simplicity we denote $f+b$ still as $f$ when doing estimates.

For convenience we denote

(3.1)

\widetilde{\Omega}=\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u).

The next lemma we need is straightforward.

Lemma 3.2.

(3.2)		$\displaystyle S_{1}(\partial\partial_{J}u)=S_{1}(\widetilde{\Omega})-S_{1}(\Omega_{h})$
(3.3)		$\displaystyle\partial\partial_{J}u=(n-1)\Omega_{h}-S_{1}(\Omega_{h})\Omega+S_{1}(\widetilde{\Omega})\Omega-(n-1)\widetilde{\Omega}.$

Proof.

From (3.1) we have

n\widetilde{\Omega}\wedge\Omega^{n-1}=n\Omega_{h}\wedge\Omega^{n-1}+\frac{n}{n-1}(S_{1}(\partial\partial_{J}u)\Omega^{n}-\partial\partial_{J}u\wedge\Omega^{n-1}).

Namely,

S_{1}(\widetilde{\Omega})=S_{1}(\Omega_{h})+\frac{1}{n-1}(nS_{1}(\partial\partial_{J}u)-S_{1}(\partial\partial_{J}u))=S_{1}(\Omega_{h})+S_{1}(\partial\partial_{J}u)

This proves (3.2), and (3.3) follows by inserting (3.2) into (3.1). ∎

Define $\Omega_{0}$ by $(n-1)!\Omega_{h}=\ast\Omega_{0}^{n-1}$ , we have the follwing

Lemma 3.3.

There exists a uniform constant $C$ such that

(3.4)

\partial\partial_{J}u\wedge(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2})\leq C\Omega^{n}

Proof.

Using (3.3) we compute

\begin{split}&\>\partial\partial_{J}u\wedge(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-1})\\ =&\>2\big{(}(n-1)\Omega_{h}-S_{1}(\Omega_{h})\Omega\big{)}\wedge\Omega_{0}^{n-1}-2\big{(}(n-1)\widetilde{\Omega}-S_{1}(\widetilde{\Omega})\Omega\big{)}\wedge\Omega_{0}^{n-1}\\ &\>+\big{(}(n-1)\Omega_{h}-S_{1}(\Omega_{h})\Omega-((n-1)\widetilde{\Omega}-S_{1}(\widetilde{\Omega})\Omega)\big{)}^{2}\wedge\Omega^{n-2}\\ =&\>2\big{(}(n-1)\Omega_{h}-S_{1}(\Omega_{h})\Omega\big{)}\wedge\Omega_{0}^{n-1}+\big{(}(n-1)\Omega_{h}-S_{1}(\Omega_{h})\Omega\big{)}^{2}\wedge\Omega^{n-2}\\ &\>-2\big{(}(n-1)\widetilde{\Omega}-S_{1}(\widetilde{\Omega})\Omega\big{)}\wedge\Omega_{0}^{n-1}-2\big{(}(n-1)\Omega_{h}-S_{1}(\Omega_{h})\Omega\big{)}\big{(}(n-1)\widetilde{\Omega}-S_{1}(\widetilde{\Omega})\Omega\big{)}\wedge\Omega^{n-2}\\ &\>+(n-1)^{2}\widetilde{\Omega}^{2}\wedge\Omega^{n-2}-2(n-1)S_{1}(\widetilde{\Omega})\widetilde{\Omega}\wedge\Omega^{n-1}+S_{1}^{2}(\widetilde{\Omega})\Omega^{n}\\ \leq&\>C\Omega^{n}-2(n-1)\widetilde{\Omega}\wedge\Omega_{0}^{n-1}+2S_{1}(\widetilde{\Omega})\Omega\wedge\Omega_{0}^{n-1}-2(n-1)^{2}\Omega_{h}\wedge\widetilde{\Omega}\wedge\Omega^{n-2}\\ &\>+2(n-1)S_{1}(\widetilde{\Omega})\Omega_{h}\wedge\Omega^{n-1}+2(n-1)S_{1}(\Omega_{h})\widetilde{\Omega}\wedge\Omega^{n-1}-2S_{1}(\Omega_{h})S_{1}(\widetilde{\Omega})\Omega^{n}\\ &\>+(n-1)^{2}\widetilde{\Omega}^{2}\wedge\Omega^{n-2}-2(n-1)S_{1}(\widetilde{\Omega})\widetilde{\Omega}\wedge\Omega^{n-1}+S_{1}^{2}(\widetilde{\Omega})\Omega^{n}.\end{split}

By definition of $S_{1}(\widetilde{\Omega})$ and $S_{n-1}(\Omega_{0})$ , we have

(3.5)

\begin{split}&\>\partial\partial_{J}u\wedge(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-1})\\ \leq&\>C\Omega^{n}-2(n-1)\widetilde{\Omega}\wedge\Omega_{0}^{n-1}+\frac{2}{n}S_{1}(\widetilde{\Omega})S_{n-1}(\Omega_{0})\Omega^{n}-2(n-1)^{2}\Omega_{h}\wedge\widetilde{\Omega}\wedge\Omega^{n-2}\\ &\>+\frac{2(n-1)}{n}S_{1}(\widetilde{\Omega})S_{1}(\Omega_{h})\Omega^{n}+\frac{2(n-1)}{n}S_{1}(\Omega_{h})S_{1}(\widetilde{\Omega})\Omega^{n}-2S_{1}(\Omega_{h})S_{1}(\widetilde{\Omega})\Omega^{n}\\ &\>+\frac{2(n-1)}{n}S_{2}(\widetilde{\Omega})\Omega^{n}-\frac{2(n-1)}{n}S_{1}^{2}(\widetilde{\Omega})\Omega^{n}+S_{1}^{2}(\widetilde{\Omega})\Omega^{n}\\ =&\>C\Omega^{n}-2(n-1)\widetilde{\Omega}\wedge\Omega_{0}^{n-1}+\frac{2}{n}S_{1}(\widetilde{\Omega})S_{n-1}(\Omega_{0})\Omega^{n}-2(n-1)^{2}\Omega_{h}\wedge\widetilde{\Omega}\wedge\Omega^{n-2}\\ &\>+\frac{2(n-2)}{n}S_{1}(\Omega_{h})S_{1}(\widetilde{\Omega})\Omega^{n}+\frac{2(n-1)}{n}S_{2}(\widetilde{\Omega})\Omega^{n}+\frac{2-n}{n}S_{1}^{2}(\widetilde{\Omega})\Omega^{n}.\end{split}

Choose local $I$ -holomorphic coordinates such that at a point, $\Omega=\sum_{i=0}^{n-1}dz^{2i}\wedge dz^{2i+1}$ and $\Omega_{0}=\sum_{i=0}^{n-1}\lambda_{i}dz^{2i}\wedge dz^{2i+1}$ with $\lambda_{i}>0$ . Since

\Omega_{h}=\frac{1}{(n-1)!}\ast\Omega_{0}^{n-1}=\sum_{i=0}^{n-1}\Lambda_{i}dz^{2i}\wedge dz^{2i+1}

where $\Lambda_{i}=\lambda_{0}\cdots\hat{\lambda_{i}}\cdots\lambda_{n-1}$ , we have $S_{1}(\Omega_{h})=S_{n-1}(\Omega_{0})=\sum_{i=0}^{n-1}\Lambda_{i}$ . Therefore

(3.6)

\frac{2}{n}S_{1}(\widetilde{\Omega})S_{n-1}(\Omega_{0})+\frac{2(n-2)}{n}S_{1}(\widetilde{\Omega})S_{1}(\Omega_{h})=\frac{2(n-1)}{n}S_{1}(\widetilde{\Omega})S_{n-1}(\Omega_{0}).

Now compute

\begin{split}2(n-1)\widetilde{\Omega}\wedge\Omega_{0}^{n-1}&=2(n-1)(n-1)!\sum_{i=0}^{n-1}\widetilde{\Omega}_{2i,2i+1}\Lambda_{i}dz^{0}\wedge\cdots\wedge dz^{2n-1}\\ 2(n-1)^{2}\Omega_{h}\wedge\widetilde{\Omega}\wedge\Omega^{n-2}&=2(n-1)(n-1)!\sum_{i=0}^{n-1}\Lambda_{i}(S_{1}(\widetilde{\Omega})-\widetilde{\Omega}_{2i,2i+1})dz^{0}\wedge\cdots\wedge dz^{2n-1}.\end{split}

Thus

(3.7)

2(n-1)\widetilde{\Omega}\wedge\Omega_{0}^{n-1}+2(n-1)^{2}\Omega_{h}\wedge\widetilde{\Omega}\wedge\Omega^{n-2}=\frac{2(n-1)}{n}S_{1}(\widetilde{\Omega})S_{n-1}(\Omega_{0})\Omega^{n}.

Combining (3.5), (3.6) and (3.7) we get

(3.8)

\partial\partial_{J}u\wedge(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2})\leq C\Omega^{n}+\frac{2(n-1)}{n}S_{2}(\widetilde{\Omega})\Omega^{n}+\frac{2-n}{n}S_{1}^{2}(\widetilde{\Omega})\Omega^{n}.

It remains to prove that the sum of the last two terms has a upper bound. The proof is analogous to that in [22], which we give here for completeness. Choose local coordinates such that at a point,

\begin{split}\Omega&=\sum_{i=0}^{n-1}dz^{2i}\wedge dz^{2i+1}\\ \widetilde{\Omega}&=\sum_{i=0}^{n-1}\mu_{i}dz^{2i}\wedge dz^{2i+1}\text{ with }0<\mu_{0}\leq\cdots\leq\mu_{n-1}.\end{split}

Then we have

\begin{split}&\>2(n-1)S_{2}(\widetilde{\Omega})+(2-n)S_{1}^{2}(\widetilde{\Omega})\\ =&\>2(n-1)\sum_{i<j}\mu_{i}\mu_{j}-(n-2)(\sum_{i=0}^{n-1}\mu_{i})^{2}\\ =&\>-(n-2)\sum_{i=0}^{n-1}\mu_{i}^{2}-2(n-2)\sum_{i<j}\mu_{i}\mu_{j}+2(n-1)\sum_{i<j}\mu_{i}\mu_{j}\\ =&\>-(n-2)\sum_{i=1}^{n-1}\mu_{i}^{2}+2\sum_{1\leq i<j\leq n-1}\mu_{i}\mu_{j}-(n-2)\mu_{0}^{2}+2\mu_{0}\sum_{i=1}^{n-1}\mu_{i}\\ \leq&\>-\sum_{1\leq i<j\leq n-1}(\mu_{i}-\mu_{j})^{2}+2\mu_{0}\sum_{i=1}^{n-1}\mu_{i}.\end{split}

We want to show this quantity has a upper bound using the equation

\mu_{0}\cdots\mu_{n-1}=e^{f}.

When $\mu_{1}<\mu_{n-1}/2$ , we have $(\mu_{1}-\mu_{n-1})^{2}\geq\frac{1}{4}\mu_{n-1}^{2}$ . Thus

-\sum_{1\leq i<j\leq n-1}(\mu_{i}-\mu_{j})^{2}+2\mu_{0}\sum_{i=1}^{n-1}\mu_{i}\leq-\frac{1}{4}\mu_{n-1}^{2}+C\mu_{n-1}\leq C^{\prime},

and the first inequality above is because $\mu_{0}$ has a uniform upper bound, being the smalest eigenvalue. When $\mu_{1}\geq\mu_{n-1}/2$ , then we have $\mu_{i}\geq\mu_{n-1}/2$ for $i=1,\cdots,n-1$ . Hence

\mu_{0}\leq\frac{C}{\mu_{1}\cdots\mu_{n-1}}\leq\frac{C2^{n-2}}{\mu_{n-1}^{n-1}}.

And in this case

-\sum_{1\leq i<j\leq n-1}(\mu_{i}-\mu_{j})^{2}+2\mu_{0}\sum_{i=1}^{n-1}\mu_{i}\leq\frac{C^{\prime}}{\mu_{n-1}^{n-1}}\mu_{n-1}=\frac{C^{\prime}}{\mu_{n-1}^{n-2}}\leq C^{\prime}.

This proves the lemma. ∎

We now establish the Cherrier-type inequality:

Lemma 3.4.

There exist uniform constants $C$ and $p_{0}$ such that for all $p\geq p_{0}$ ,

(3.9)

\int_{M}|\partial e^{-\frac{pu}{2}}|^{2}_{g}\Omega^{n}\wedge\overline{\Omega}{\mathstrut}^{n}\leq Cp\int_{M}e^{-pu}\Omega^{n}\wedge\overline{\Omega}{\mathstrut}^{n}.

Proof.

By Lemma 3.3 we have

\mathcal{I}:=\int_{M}e^{-pu}\partial\partial_{J}u\wedge(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2})\wedge\overline{\Omega}{\mathstrut}^{n}\leq C\int_{M}e^{-pu}\Omega^{n}\wedge\overline{\Omega}{\mathstrut}^{n}.

Interating by parts, we have

	$\displaystyle\mathcal{I}=$	$\displaystyle-\int_{M}\partial e^{-pu}\wedge\partial_{J}u\wedge\Big{(}2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2}\Big{)}\wedge{\overline{\Omega}}^{n}$
		$\displaystyle+\int_{M}e^{-pu}\partial_{J}u\wedge\partial\Big{(}(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2})\wedge{\overline{\Omega}}^{n}\Big{)}$
	$\displaystyle=$	$\displaystyle p\int_{M}e^{-pu}\partial u\wedge\partial_{J}u\wedge\Big{(}2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2}\Big{)}\wedge{\overline{\Omega}}^{n}$
		$\displaystyle+\int_{M}e^{-pu}\partial_{J}u\wedge\Big{(}(2\partial\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\partial\Omega^{n-2})\wedge{\overline{\Omega}}^{n}+(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2})\wedge\partial{\overline{\Omega}}^{n}\Big{)}$
	$\displaystyle=$	$\displaystyle\mathcal{I}_{1}+\mathcal{I}_{2}$

Since $\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2}>0$ (see (2.10)), we obtain

\displaystyle\mathcal{I}_{1}\geq p\int_{M}e^{-pu}\partial u\wedge\partial_{J}u\wedge\Omega_{0}^{n-1}\wedge{\overline{\Omega}}^{n}\geq c_{0}p\int_{M}e^{-pu}\partial u\wedge\partial_{J}u\wedge\Omega^{n-1}\wedge{\overline{\Omega}}^{n},

where we use $\Omega_{0}\geq c_{0}^{\frac{1}{n-1}}\Omega$ for a positive constant $c_{0}$ .

Next we estimate $\mathcal{I}_{2}$ . Indeed, we have

	$\displaystyle\mathcal{I}_{2}=$	$\displaystyle-\frac{1}{p}\int_{M}{\partial_{J}e^{-pu}\wedge\Big{(}(2\partial\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\partial\Omega^{n-2})\wedge{\overline{\Omega}}^{n}+(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2})\wedge\partial{\overline{\Omega}}^{n}\Big{)}}$
	$\displaystyle=$	$\displaystyle\frac{1}{p}\int_{M}{e^{-pu}\Big{(}(2\partial_{J}\partial\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\partial_{J}\partial\Omega^{n-2})\wedge{\overline{\Omega}}^{n}-(2\partial\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\partial\Omega^{n-2})\wedge\partial_{J}{\overline{\Omega}}^{n}\Big{)}}$
		$\displaystyle+\frac{1}{p}\int_{M}{e^{-pu}\Big{(}(2\partial_{J}\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\partial_{J}\Omega^{n-2})\wedge\partial{\overline{\Omega}}^{n}+(2\Omega_{0}^{n-1}+\partial\partial_{J}u\wedge\Omega^{n-2})\wedge{\partial_{J}\partial\overline{\Omega}}^{n}\Big{)}}$
	$\displaystyle=$	$\displaystyle\frac{1}{p}\int_{M}{e^{-pu}\partial\partial_{J}u\wedge\Big{(}\partial_{J}\partial\Omega^{n-2}\wedge{\overline{\Omega}}^{n}-\partial\Omega^{n-2}\wedge\partial_{J}{\overline{\Omega}}^{n}+\partial_{J}\Omega^{n-2}\wedge\partial{\overline{\Omega}}^{n}+\Omega^{n-2}\wedge\partial_{J}\partial{\overline{\Omega}}^{n}\Big{)}}$
		$\displaystyle+\frac{1}{p}\int_{M}{e^{-pu}\Big{(}2\partial_{J}\partial\Omega_{0}^{n-1}\wedge{\overline{\Omega}}^{n}-2\partial\Omega_{0}^{n-1}\wedge\partial_{J}{\overline{\Omega}}^{n}+2\partial_{J}\Omega_{0}^{n-1}\wedge\partial{\overline{\Omega}}^{n}+2\Omega_{0}^{n-1}\wedge\partial_{J}\partial{\overline{\Omega}}^{n}\Big{)}}$
	$\displaystyle=$	$\displaystyle\mathcal{I}_{21}+\mathcal{I}_{22}.$

$\mathcal{I}_{22}$ has the following estimate:

\mathcal{I}_{22}\geq-Cp^{-1}\int_{M}{e^{-pu}\Omega^{n}\wedge\overline{\Omega}^{n}}.

Integrating by parts, we have

	$\displaystyle\mathcal{I}_{21}=$	$\displaystyle\int_{M}{e^{-pu}\partial u\wedge\partial_{J}u\wedge\Big{(}\partial_{J}\partial\Omega^{n-2}\wedge{\overline{\Omega}}^{n}-\partial\Omega^{n-2}\wedge\partial_{J}{\overline{\Omega}}^{n}+\partial_{J}\Omega^{n-2}\wedge\partial{\overline{\Omega}}^{n}+\Omega^{n-2}\wedge\partial_{J}\partial{\overline{\Omega}}^{n}\Big{)}}$
	$\displaystyle+$	$\displaystyle\frac{1}{p}\int_{M}{e^{-pu}\partial_{J}u\wedge\Big{(}\partial_{J}\partial\Omega^{n-2}\wedge\partial{\overline{\Omega}}^{n}+\partial\Omega^{n-2}\wedge\partial\partial_{J}{\overline{\Omega}}^{n}+\partial\partial_{J}\Omega^{n-2}\wedge\partial{\overline{\Omega}}^{n}+\partial\Omega^{n-2}\wedge\partial_{J}\partial{\overline{\Omega}}^{n}\Big{)}}$
	$\displaystyle\geq$	$\displaystyle-C\int_{M}{e^{-pu}\partial u\wedge\partial_{J}u\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}}.$

Therefore we obtain

	$\displaystyle\mathcal{I}\geq(c_{0}p-C)\int_{M}{e^{-pu}\partial u\wedge\partial_{J}u\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}}-\frac{C}{p}\int_{M}{e^{-pu}\Omega^{n}\wedge\overline{\Omega}^{n}}$
	$\displaystyle\geq\frac{c_{0}p}{2}\int_{M}{e^{-pu}\partial u\wedge\partial_{J}u\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}}-\frac{C}{p}\int_{M}{e^{-pu}\Omega^{n}\wedge\overline{\Omega}^{n}}.$

Take $p_{0}=(2c_{0})^{-1}C$ , then for all $p\geq p_{0}$ ,

\frac{1}{p}\int_{M}\partial e^{-\frac{pu}{2}}\wedge\partial_{J}e^{-\frac{pu}{2}}\wedge\Omega^{n-1}\wedge\overline{\Omega}{\mathstrut}^{n}\leq C\int_{M}e^{-pu}\Omega\wedge\overline{\Omega}{\mathstrut}^{n}.

This proves the lemma. ∎

Proof of Theorem 3.1.

From Lemma 3.4, we can prove the $C^{0}$ estimate using similar arguments as that in [12] and [21, 20, 23] by regarding $M$ as a Hermitian manifold $(M,I,g)$ . For completeness, we sketch the proof here.

By [20], the Cherrier-type inequality (3.9) implies

e^{-p_{0}\inf\limits_{M}u}\leq C\int_{M}e^{-p_{0}u}\omega_{I}^{2n}.

Then by [12] or [21] there exist uniform constants $C_{1}$ and $\delta>0$ such that

|\{u\leq\inf_{M}u+C_{1}\}|_{\omega_{I}}\geq\delta.

On the other hand, from $\sup_{M}u=0$ and $\Delta_{\omega_{I}}u=2S_{1}(\partial\partial_{J}u)\geq-2S_{1}(\Omega_{h})$ (see (3.2)), one can show that (see [23])

\int_{M}(-u)\omega_{I}^{2n}\leq C_{2}.

Then we have

-\delta\inf_{M}u\leq\int_{\{u\leq\inf\limits_{M}u+C_{1}\}}(-u+C_{1})\leq C.

This finishes the proof. ∎

4. $C^{1}$ Estimate

Theorem 4.1.

Let $u$ be a solution as in Theorem 1.1. Then there exists a constant $C$ depending only on the fixed data $(I,J,K,g,\Omega,\Omega_{h})$ and $f$ such that

(4.1)

|du|_{g}\leq C.

Proof.

A simple computation in local coordinates shows that

n\partial u\wedge\partial_{J}u\wedge\Omega^{n-1}=\frac{1}{4}|du|_{g}^{2}\Omega^{n}.

Define

\beta\coloneqq\frac{1}{4}|du|_{g}^{2}.

Following [6], we consider

G=\log\beta-\varphi\circ u

where $\varphi$ is a function to be determined. Suppose $G$ attain its maximum at $p$ , and from now on we compute at the point $p$ using the normal coordinates around $p$ (see Remark 2.9).

\begin{split}\partial G&=\frac{\partial\beta}{\beta}-\varphi^{\prime}\partial u=0;\\ \partial_{J}G&=\frac{\partial_{J}\beta}{\beta}-\varphi^{\prime}\partial_{J}u=0;\\ \partial\partial_{J}G&=\frac{\partial\partial_{J}\beta}{\beta}-\frac{\partial\beta\wedge\partial_{J}\beta}{\beta^{2}}-\varphi^{\prime\prime}\partial u\wedge\partial_{J}u-\varphi^{\prime}\partial\partial_{J}u\\ &=\frac{\partial\partial_{J}\beta}{\beta}-((\varphi^{\prime})^{2}+\varphi^{\prime\prime})\partial u\wedge\partial_{J}u-\varphi^{\prime}\partial\partial_{J}u.\end{split}

Let

(4.2)

A=S_{n-1}(\widetilde{\Omega})\Omega^{n-1}-\widetilde{\Omega}^{n-1},

where $\widetilde{\Omega}$ is as in the last section. Computing in normal coordinates shows

A=(n-1)!\sum_{i=0}^{n-1}(\sum_{j\neq i}\frac{\widetilde{\Omega}_{01}\cdots\widetilde{\Omega}_{2n-2\>2n-1}}{\widetilde{\Omega}_{2j2j+1}})dz^{0}\wedge dz^{1}\wedge\cdots\wedge\widehat{dz^{2i}}\wedge\widehat{dz^{2i+1}}\wedge\cdots\wedge dz^{2n-2}\wedge dz^{2n-1}.

Thus $A$ is positive, and we have at point $p$

(4.3)

\begin{split}0&\geq\frac{\partial\partial_{J}G\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}\\ &=\frac{\partial\partial_{J}\beta\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\beta\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}-((\varphi^{\prime})^{2}+\varphi^{\prime\prime})\frac{\partial u\wedge\partial_{J}u\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}-\varphi^{\prime}\frac{\partial\partial_{J}u\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}.\end{split}

We need to compute $\partial\partial_{J}\beta$ . By definition of $\beta$ we have

\beta\overline{\Omega}{\mathstrut}^{n}=n\overline{\partial}u\wedge\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}.

Taking $\partial_{J}$ of both sides and noticing $\partial_{J}\Omega=0$ since $\Omega$ is hyperKähler, we get

\partial_{J}\beta\wedge\overline{\Omega}{\mathstrut}^{n}=n\partial_{J}\overline{\partial}u\wedge\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}-n\overline{\partial}u\wedge\partial_{J}\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}.

Then taking $\partial$ of both sides, we get

\begin{split}\partial\partial_{J}\beta\wedge\overline{\Omega}{\mathstrut}^{n}=&n\partial\partial_{J}\overline{\partial}u\wedge\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}+n\partial_{J}\overline{\partial}u\wedge\partial\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}\\ &-n\partial\overline{\partial}u\wedge\partial_{J}\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}+n\overline{\partial}u\wedge\partial\partial_{J}\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}.\end{split}

From the equation

(4.4)

\widetilde{\Omega}^{n}=e^{f}\Omega^{n},

by taking $\overline{\partial}$ we obtain

A\wedge n\overline{\partial}\partial\partial_{J}u=(n-1)(-n\widetilde{\Omega}^{n-1}\wedge\overline{\partial}\Omega_{h}+\overline{\partial}e^{f}\wedge\Omega^{n}),

and by taking $\overline{\partial_{J}}$ we obtain

A\wedge n\overline{\partial_{J}}\partial\partial_{J}u=(n-1)(-n\widetilde{\Omega}^{n-1}\wedge\overline{\partial_{J}}\Omega_{h}+\overline{\partial_{J}}e^{f}\wedge\Omega^{n}).

Thus we have for the first term of (4.3)

(4.5)

\partial\partial_{J}\beta\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}=I_{1}+I_{2}+n\partial_{J}\overline{\partial}u\wedge\partial\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}\wedge A-n\partial\overline{\partial}u\wedge\partial_{J}\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}\wedge A

where

\begin{split}I_{1}&=(n-1)(-n\widetilde{\Omega}^{n-1}\wedge\overline{\partial}\Omega_{h}+\overline{\partial}e^{f}\wedge\Omega^{n})\wedge\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1},\\ I_{2}&=(n-1)(n\widetilde{\Omega}^{n-1}\wedge\overline{\partial_{J}}\Omega_{h}-\overline{\partial_{J}}e^{f}\wedge\Omega^{n})\wedge\overline{\partial}u\wedge\overline{\Omega}{\mathstrut}^{n-1}.\end{split}

By direct computation,

\begin{split}\partial_{J}\overline{\partial}u&=\sum u_{\overline{ji}}J^{-1}d\overline{z^{i}}\wedge d\overline{z^{j}};\\ \partial\overline{\partial_{J}}u&=\sum u_{ij}dz^{j}\wedge J^{-1}dz^{i};\\ \partial\overline{\partial}u&=\sum u_{i\overline{j}}dz^{i}\wedge d\overline{z^{j}};\\ \partial_{J}\overline{\partial_{J}}u&=\sum u_{i\overline{j}}J^{-1}d\overline{z^{j}}\wedge J^{-1}dz^{i};\end{split}

Thus the third term of (4.5) become

(4.6)

n\partial_{J}\overline{\partial}u\wedge\partial\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}\wedge A=\frac{1}{n}\sum_{k=0}^{n-1}\sum_{j=0}^{2n-1}(\sum_{i\neq k}\frac{1}{\widetilde{\Omega}_{2i2i+1}})(|u_{2kj}|^{2}+|u_{2k+1j}|^{2})\widetilde{\Omega}^{n}\wedge\overline{\Omega}{\mathstrut}^{n};

and the forth term

(4.7)

-n\partial\overline{\partial}u\wedge\partial_{J}\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}\wedge A=\frac{1}{n}\sum_{k=0}^{n-1}\sum_{j=0}^{2n-1}(\sum_{i\neq k}\frac{1}{\widetilde{\Omega}_{2i2i+1}})(|u_{2k\overline{j}}|^{2}+|u_{2k+1\overline{j}}|^{2})\widetilde{\Omega}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}.

For $I_{1}$ and $I_{2}$ we have

(4.8)

\begin{split}\frac{1}{n-1}I_{1}&=-n\widetilde{\Omega}^{n-1}\wedge\overline{\partial}\Omega_{h}\wedge\overline{\partial_{J}}u\wedge\overline{\Omega}{\mathstrut}^{n-1}-\overline{\partial_{J}}u\wedge\overline{\partial}e^{f}\wedge\Omega^{n}\wedge\overline{\Omega}{\mathstrut}^{n-1}\\ &=-\frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=0}^{2n-1}\frac{(\Omega_{h})_{2i2i+1,\overline{j}}u_{j}}{\widetilde{\Omega}_{2i2i+1}}\widetilde{\Omega}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}+\frac{1}{n}\sum_{j=0}^{2n-1}\frac{u_{j}(e^{f})_{\overline{j}}}{e^{f}}\widetilde{\Omega}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}\end{split}

and

(4.9)

\begin{split}\frac{1}{n-1}I_{2}&=n\widetilde{\Omega}^{n-1}\wedge\overline{\partial_{J}}\Omega_{h}\wedge\overline{\partial}u\wedge\overline{\Omega}{\mathstrut}^{n-1}+\overline{\partial}u\wedge\overline{\partial_{J}}e^{f}\wedge\Omega^{n}\wedge\overline{\Omega}{\mathstrut}^{n-1}\\ &=-\frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=0}^{2n-1}\frac{(\overline{\Omega}{\mathstrut}_{h})_{2i2i+1,j}u_{\overline{j}}}{\widetilde{\Omega}_{2i2i+1}}\widetilde{\Omega}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}+\frac{1}{n}\sum_{j=0}^{2n-1}\frac{u_{\overline{j}}(e^{f})_{j}}{e^{f}}\widetilde{\Omega}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}.\end{split}

Combining (4.8), (4.9), (4.6), (4.7) we obtain estimate of (4.5)

(4.10)

\begin{split}\frac{\partial\partial_{J}\beta\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\beta\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}&=-\frac{1}{n\beta}\sum_{i=0}^{n-1}\sum_{j=0}^{2n-1}\frac{(\Omega_{h})_{2i2i+1,\overline{j}}u_{j}+(\overline{\Omega}{\mathstrut}_{h})_{2i2i+1,j}u_{\overline{j}}}{\widetilde{\Omega}_{2i2i+1}}+\frac{1}{n\beta}\sum_{j=0}^{2n-1}\frac{u_{j}(e^{f})_{\overline{j}}+u_{\overline{j}}(e^{f})_{j}}{e^{f}}\\ &+\frac{1}{n\beta}\sum_{k=0}^{n-1}\sum_{j=0}^{2n-1}\sum_{i\neq k}\frac{|u_{2kj}|^{2}+|u_{2k+1j}|^{2}}{\widetilde{\Omega}_{2i2i+1}}+\frac{1}{n\beta}\sum_{k=0}^{n-1}\sum_{j=0}^{2n-1}\sum_{i\neq k}\frac{|u_{2k\overline{j}}|^{2}+|u_{2k+1\overline{j}}|^{2}}{\widetilde{\Omega}_{2i2i+1}}.\end{split}

Again by direct computation, the second term of (4.3) is

(4.11)

\partial u\wedge\partial_{J}u\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}=\frac{1}{n}\sum_{i=0}^{n-1}(\sum_{k\neq i}\frac{1}{\widetilde{\Omega}_{2k2k+1}})(|u_{2i}|^{2}+|u_{2i+1}|^{2})\widetilde{\Omega}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}.

For the third term of (4.3), we compute

(4.12)

\begin{split}\partial\partial_{J}u\wedge A&=\partial\partial_{J}u\wedge(\frac{n\widetilde{\Omega}^{n-1}\wedge\Omega}{\Omega^{n}}\Omega^{n-1}-\widetilde{\Omega}^{n-1})\\ &=(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u)\wedge\widetilde{\Omega}^{n-1}\\ &=(n-1)(\widetilde{\Omega}^{n}-\Omega_{h}\wedge\widetilde{\Omega}^{n-1}).\end{split}

By compactness of $M$ , there exists $\epsilon>0$ such that $\Omega_{h}\geq\epsilon\Omega$ , we obtain

(4.13)

\begin{split}-\varphi^{\prime}\frac{\partial\partial_{J}u\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}&=-(n-1)\varphi^{\prime}+(n-1)\varphi^{\prime}\frac{\Omega_{h}\wedge\widetilde{\Omega}^{n-1}\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}\\ &\geq-(n-1)\varphi^{\prime}+\frac{\epsilon(n-1)\varphi^{\prime}}{n}\sum_{i=0}^{n-1}\frac{1}{\widetilde{\Omega}_{2i2i+1}}.\end{split}

We may assume $\beta\gg 1$ otherwise we are finished. The inequality (4.3) become

(4.14)

\begin{split}0\geq&\frac{n-1}{n\beta e^{f}}\sum_{i=0}^{2n-1}(u_{i}(e^{f})_{\overline{i}}+u_{\overline{i}}(e^{f})_{i})\\ &-\frac{(\varphi^{\prime})^{2}+\varphi^{\prime\prime}}{n}\sum_{i=0}^{n-1}(\sum_{k\neq i}\frac{1}{\widetilde{\Omega}_{2k2k+1}})(|u_{2i}|^{2}+|u_{2i+1}|^{2})\\ &-(n-1)\varphi^{\prime}+\frac{n-1}{n}(\epsilon\varphi^{\prime}-C_{1}\frac{\sum u_{j}}{\beta}-C_{2}\frac{\sum u_{\overline{j}}}{\beta})\sum_{i=0}^{n-1}\frac{1}{\widetilde{\Omega}_{2i2i+1}}.\end{split}

The first term is bounded from below. Now we take

(4.15)

\varphi(t)=\frac{\log(2t+C_{0})}{2}.

where $C_{0}$ is determined by $C^{0}$ estimate, and rewrite (4.14) as

(4.16)

C_{3}\geq C_{4}\sum_{i=0}^{n-1}(\sum_{k\neq i}\frac{1}{\widetilde{\Omega}_{2k2k+1}})(|u_{2i}|^{2}+|u_{2i+1}|^{2})+C_{5}\sum_{i=0}^{n-1}\frac{1}{\widetilde{\Omega}_{2i2i+1}}.

Thus for any fixed $i$

\widetilde{\Omega}_{2i2i+1}\geq\frac{C_{5}}{C_{3}}\geq C.

By equation (4.4) we also have

\frac{1}{\widetilde{\Omega}_{2i2i+1}}=e^{-f}\prod_{j\neq i}\widetilde{\Omega}_{2j2j+1}\geq\frac{C^{n-1}}{\sup_{M}e^{f}}.

From the bound on all $\widetilde{\Omega}_{2i2i+1}$ , we obtain the bound on $\beta$ by (4.16). ∎

5. Bound on $\partial\partial_{J}u$

Theorem 5.1.

Let $u$ be a solution as in Theorem 1.1. Then there exists a constant $C$ depending only on the fixed data $(I,J,K,g,\Omega,\Omega_{h})$ and $f$ such that

(5.1)

|\partial\partial_{J}u|_{g}\leq C.

Proof.

For simplicity denote

\eta=S_{1}(\partial\partial_{J}u).

Consider the function

G=\log\eta-\varphi\circ u

where the function $\varphi$ is as in the previous section. We compute at a maximum point $p$ of $G$ using the normal coordinates around $p$ (see Remark 2.9). We have

\begin{split}\partial G&=\frac{\partial\eta}{\eta}-\varphi^{\prime}\partial u=0;\\ \partial_{J}G&=\frac{\partial_{J}\eta}{\eta}-\varphi^{\prime}\partial_{J}u=0;\\ \partial\partial_{J}G&=\frac{\partial\partial_{J}\eta}{\eta}-((\varphi^{\prime})^{2}+\varphi^{\prime\prime})\partial u\wedge\partial_{J}u-\varphi^{\prime}\partial\partial_{J}u.\end{split}

Let $A$ be as before (see (4.2)), then at point $p$ we have

(5.2)

\begin{split}0&\geq\frac{\partial\partial_{J}G\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}\\ &=\frac{\partial\partial_{J}\eta\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\eta\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}-((\varphi^{\prime})^{2}+\varphi^{\prime\prime})\frac{\partial u\wedge\partial_{J}u\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}-\varphi^{\prime}\frac{\partial\partial_{J}u\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}.\end{split}

The second and the third term were dealt with in the previous section. We now focus on $\partial\partial_{J}\eta$ in the first term.

By definition $\eta$ is real, and

\eta\overline{\Omega}{\mathstrut}^{n}=n\bar{\partial}\bar{\partial}_{J}u\wedge\overline{\Omega}{\mathstrut}^{n-1}.

Under the hyperKähler condition $\mathrm{d}\>\!\Omega=0$ , differentiating twice the above equation gives

(5.3)

\partial\partial_{J}\eta\wedge\overline{\Omega}{\mathstrut}^{n}=n\partial\partial_{J}\bar{\partial}\bar{\partial}_{J}u\wedge\overline{\Omega}{\mathstrut}^{n-1}=n\bar{\partial}\bar{\partial}_{J}\partial\partial_{J}u\wedge\overline{\Omega}{\mathstrut}^{n-1}

The last equality above is due to Lemma 2.1.

We know that (recall (3.3))

\partial\partial_{J}u=(n-1)\Omega_{h}-S_{1}(\Omega_{h})\Omega+S_{1}(\widetilde{\Omega})\Omega-(n-1)\widetilde{\Omega}.

Thus

(5.4)

\bar{\partial}\bar{\partial}_{J}\partial\partial_{J}u=(n-1)\bar{\partial}\bar{\partial}_{J}\Omega_{h}-\bar{\partial}\bar{\partial}_{J}S_{1}(\Omega_{h})\wedge\Omega+\bar{\partial}\bar{\partial}_{J}S_{1}(\widetilde{\Omega})\wedge\Omega-(n-1)\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}.

Here we again used the hyperKähler condition on $\Omega$ . Now we have

(5.5)

\begin{split}\partial\partial_{J}\eta\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}&=nA\wedge\bar{\partial}\bar{\partial}_{J}\partial\partial_{J}u\wedge\overline{\Omega}{\mathstrut}^{n-1}\\ &=n(n-1)A\wedge\bar{\partial}\bar{\partial}_{J}\Omega_{h}\wedge\overline{\Omega}{\mathstrut}^{n-1}-n\bar{\partial}\bar{\partial}_{J}S_{1}(\Omega_{h})\wedge A\wedge\Omega\wedge\overline{\Omega}{\mathstrut}^{n-1}\\ &\quad+n\bar{\partial}\bar{\partial}_{J}S_{1}(\widetilde{\Omega})\wedge A\wedge\Omega\wedge\overline{\Omega}{\mathstrut}^{n-1}-n(n-1)A\wedge\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}\wedge\overline{\Omega}{\mathstrut}^{n-1}\end{split}

Notice that

A\wedge\Omega=S_{n-1}(\widetilde{\Omega})\Omega^{n}-\widetilde{\Omega}^{n-1}\wedge\Omega=\frac{n-1}{n}S_{n-1}(\widetilde{\Omega})\Omega^{n}

and

\bar{\partial}\bar{\partial}_{J}S_{1}(\widetilde{\Omega})\wedge\Omega^{n}=n\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}\wedge\Omega^{n-1}.

The third term of (5.5) becomes

\begin{split}\bar{\partial}\bar{\partial}_{J}S_{1}(\widetilde{\Omega})\wedge A\wedge\Omega\wedge\overline{\Omega}{\mathstrut}^{n-1}&=\bar{\partial}\bar{\partial}_{J}S_{1}(\widetilde{\Omega})\wedge(\Omega^{n}\cdot\frac{n-1}{n}S_{n-1}(\widetilde{\Omega}))\wedge\overline{\Omega}{\mathstrut}^{n-1}\\ &=(n-1)S_{n-1}(\widetilde{\Omega})\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}\wedge\Omega^{n-1}\wedge\overline{\Omega}{\mathstrut}^{n-1}.\end{split}

The forth term is

A\wedge\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}\wedge\overline{\Omega}{\mathstrut}^{n-1}=S_{n-1}(\widetilde{\Omega})\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}\wedge\Omega^{n-1}\wedge\overline{\Omega}{\mathstrut}^{n-1}-\widetilde{\Omega}^{n-1}\wedge\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}\wedge\overline{\Omega}{\mathstrut}^{n-1}.

The first two terms of (5.5) are similar and we get

\partial\partial_{J}\eta\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}=n(n-1)\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}\wedge\widetilde{\Omega}^{n-1}\wedge\overline{\Omega}{\mathstrut}^{n-1}-n(n-1)\bar{\partial}\bar{\partial}_{J}\Omega_{h}\wedge\widetilde{\Omega}^{n-1}\wedge\overline{\Omega}{\mathstrut}^{n-1}

and

(5.6)

\begin{split}\frac{\partial\partial_{J}\eta\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\eta\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}&=n(n-1)\frac{\bar{\partial}\bar{\partial}_{J}\widetilde{\Omega}\wedge\widetilde{\Omega}^{n-1}\wedge\overline{\Omega}{\mathstrut}^{n-1}}{\eta\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}-n(n-1)\frac{\bar{\partial}\bar{\partial}_{J}\Omega_{h}\wedge\widetilde{\Omega}^{n-1}\wedge\overline{\Omega}{\mathstrut}^{n-1}}{\eta\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}\\ &=\frac{n-1}{\eta n}\sum_{i=0}^{n-1}\sum_{p=0}^{2n-1}\frac{\widetilde{\Omega}_{2i2i+1,p\bar{p}}}{\widetilde{\Omega}_{2i2i+1}}-\frac{n-1}{\eta n}\sum_{i=0}^{n-1}\sum_{p=0}^{2n-1}\frac{(\Omega_{h})_{2i2i+1,p\bar{p}}}{\widetilde{\Omega}_{2i2i+1}}\\ &\geq\frac{n-1}{\eta n}\sum_{i=0}^{n-1}\sum_{p=0}^{2n-1}\frac{\widetilde{\Omega}_{2i2i+1,p\bar{p}}}{\widetilde{\Omega}_{2i2i+1}}-\frac{C_{1}}{\eta}\sum_{i=0}^{n-1}\frac{1}{\widetilde{\Omega}_{2i2i+1}}.\end{split}

We now rewrite the right hand side of (5.6) using the equation

(5.7)

\text{Pf}(\widetilde{\Omega}_{ij})=e^{f}\text{Pf}(\Omega_{ij}).

Take logarithm of both sides

(5.8)

\log\text{Pf}(\widetilde{\Omega}_{ij})=f+\log\text{Pf}(\Omega_{ij}).

Since $\Omega^{n}=\text{Pf}(\Omega_{ij})dz^{0}\wedge\cdots\wedge dz^{2n-1}$ and $\bar{\partial}\Omega=0$ , we have $\bar{\partial}\text{Pf}(\Omega)=0$ . Taking $\bar{\partial}$ of (5.8), since $\text{Pf}(\widetilde{\Omega}_{ij})^{2}=\det(\widetilde{\Omega}_{ij})$ , we get

(5.9)

\frac{1}{2}\sum\widetilde{\Omega}^{ij}\widetilde{\Omega}_{ji,\bar{p}}=f_{\bar{p}}.

Taking $\partial$ of both sides we obtain

(5.10)

\frac{1}{2}\sum\widetilde{\Omega}^{ij}\widetilde{\Omega}_{ji,\bar{p}p}=\frac{1}{2}\sum\widetilde{\Omega}^{ik}\widetilde{\Omega}_{kl,p}\widetilde{\Omega}^{lj}\widetilde{\Omega}_{ji,\bar{p}}+f_{p\bar{p}}.

Writing in local coordinates, the left hand side of (5.10) is

(5.11)

\frac{1}{2}\sum\widetilde{\Omega}^{2i2i+1}\widetilde{\Omega}_{2i+12i,p\bar{p}}+\frac{1}{2}\sum\widetilde{\Omega}^{2i+12i}\widetilde{\Omega}_{2i2i+1,p\bar{p}}=\sum\frac{\widetilde{\Omega}_{2i2i+1,p\bar{p}}}{\widetilde{\Omega}_{2i2i+1}}.

We claim that the first term of the right hand side of (5.10) is positive, i.e.

(5.12)

\sum\widetilde{\Omega}^{ik}\widetilde{\Omega}_{kl,p}\widetilde{\Omega}^{lj}\widetilde{\Omega}_{ji,\bar{p}}\geq 0.

Indeed, in canonical coordinates,

\begin{split}\sum\widetilde{\Omega}^{ik}\widetilde{\Omega}_{kl,p}\widetilde{\Omega}^{lj}\widetilde{\Omega}_{ji,\bar{p}}&=\widetilde{\Omega}^{2i2i+1}(\widetilde{\Omega}^{2j2j+1}\widetilde{\Omega}_{2i+12j,p}\widetilde{\Omega}_{2j+12i,\bar{p}}+\widetilde{\Omega}^{2j+12j}\widetilde{\Omega}_{2i+12j+1,p}\widetilde{\Omega}_{2j2i,\bar{p}})\\ &\quad+\widetilde{\Omega}^{2i+12i}(\widetilde{\Omega}^{2j2j+1}\widetilde{\Omega}_{2i2j,p}\widetilde{\Omega}_{2j+12i+1,\bar{p}}+\widetilde{\Omega}^{2j+12j}\widetilde{\Omega}_{2i2j+1,p}\widetilde{\Omega}_{2j2i+1,\bar{p}})\\ &=\sum\frac{\widetilde{\Omega}_{2i+12j,p}\widetilde{\Omega}_{2j+12i,\bar{p}}+\widetilde{\Omega}_{2i2j+1,p}\widetilde{\Omega}_{2j2i+1,\bar{p}}}{\widetilde{\Omega}_{2i2i+1}\widetilde{\Omega}_{2j2j+1}}\\ &\quad-\sum\frac{\widetilde{\Omega}_{2i+12j+1,p}\widetilde{\Omega}_{2j2i,\bar{p}}+\widetilde{\Omega}_{2i2j,p}\widetilde{\Omega}_{2j+12i+1,\bar{p}}}{\widetilde{\Omega}_{2i2i+1}\widetilde{\Omega}_{2j2j+1}}\end{split}

Since $\widetilde{\Omega}$ is $J$ -real, using relation (2.12) we see that

(5.13)

\widetilde{\Omega}^{ik}\widetilde{\Omega}_{klp}\widetilde{\Omega}^{lj}\widetilde{\Omega}_{ji\bar{p}}=\sum\frac{|\widetilde{\Omega}_{2i+12j,p}|^{2}+|\widetilde{\Omega}_{2i2j+1,p}|^{2}+|\widetilde{\Omega}_{2i+12j+1,p}|^{2}+|\widetilde{\Omega}_{2i2j,p}|^{2}}{\widetilde{\Omega}_{2i2i+1}\widetilde{\Omega}_{2j2j+1}}

therefore (5.12) holds. By (5.6), (5.10), (5.11) and (5.12) we have

(5.14)

\frac{\partial\partial_{J}\eta\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\eta\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}\geq\frac{n-1}{2\eta n}\Delta_{I,g}f-\frac{C_{1}}{\eta}\sum_{i=0}^{n-1}\frac{1}{\widetilde{\Omega}_{2i2i+1}}.

By (4.11) and (4.13), the inequality (5.2) now becomes

(5.15)

\begin{split}0\geq&\frac{n-1}{2\eta n}\Delta_{I,g}f-\frac{(\varphi^{\prime})^{2}+\varphi^{\prime\prime}}{n}\sum_{i=0}^{n-1}(\sum_{k\neq i}\frac{1}{\widetilde{\Omega}_{2k2k+1}})(|u_{2i}|^{2}+|u_{2i+1}|^{2})\\ &-(n-1)\varphi^{\prime}+\left(\frac{\epsilon(n-1)\varphi^{\prime}}{n}-\frac{C_{1}}{\eta}\right)\sum_{i=0}^{n-1}\frac{1}{\widetilde{\Omega}_{2i2i+1}}.\end{split}

Assuming $\eta\gg 1$ , we obtain from (5.15)

(5.16)

C_{2}\geq C_{3}\sum_{i=0}^{n-1}\frac{1}{\widetilde{\Omega}_{2i2i+1}}

and hence all $\widetilde{\Omega}_{2i2i+1}$ are uniformly bounded. Since $\eta=S_{1}(\partial\partial_{J}u)=S_{1}(\widetilde{\Omega})-S_{1}(\Omega_{h})$ , we can therefore obtain a unform bound on $\eta$ .

∎

6. $C^{2}$ Estimate

Theorem 6.1.

Let $u$ be a solution as in Theorem 1.1. Then there exists a constant $C$ depending only on the fixed data $(I,J,K,g,\Omega,\Omega_{h})$ and $f$ such that

(6.1)

|\nabla^{2}u|_{g}\leq C.

Proof.

Since the sum of eigenvalues of $\nabla^{2}u$ is bounded below by

\frac{1}{2}\Delta_{I,g}u=S_{1}(\partial\partial_{J}u)=S_{1}(\widetilde{\Omega})-S_{1}(\Omega_{h})\geq-S_{1}(\Omega_{h}),

it is sufficient to show that the maximum eigenvalue is bounded from above. Define a function on $M$ as in [7]

\lambda(x)=\sup_{X\in S(T_{x}M)}(\nabla^{2}u)(X,X)

where $S(T_{x}M)$ denotes unit tangent vectors at $x$ .

Consider the function

G=\lambda+\frac{1}{4}|du|_{g}^{2}.

Since we have obtained $C^{1}$ estimate, it is sufficient to estimate $G$ at a maximum point $p\in M$ . In the normal coordinates around $p$ we introduce real coordinates

(6.2)

z^{j}=t_{j}+it_{2n+j},\quad j=0,\cdots,2n-1,

and compute

(6.3)

\nabla^{2}u=\nabla(u_{t_{j}}dt_{j})=u_{t_{i}t_{j}}dt_{i}\otimes dt_{j}-\Gamma^{k}_{ji}u_{t_{j}}dt_{i}\otimes dt_{k},

where $\Gamma^{k}_{ji}$ is the Christoffel symbol of $\nabla$ with respect to $\{\frac{\partial}{\partial t_{j}}\}^{4n-1}_{i=0}$ . Suppose

X(p)=\sum_{j=0}^{4n-1}X^{j}(p)\frac{\partial}{\partial t_{j}}(p)

is the vector realizing the supremum of $\nabla^{2}u$ at $p$ , and we extend it to a constant vector field $X$ near $p$ , i.e.

X=\sum_{j=0}^{4n-1}X^{j}(p)\frac{\partial}{\partial t_{j}}.

Then define in a sufficiently small neighbourhood,

\begin{split}\tilde{\lambda}&=\nabla^{2}u(X,X)\\ \tilde{G}&=\tilde{\lambda}+\frac{1}{4}|du|_{g}^{2}.\end{split}

Notice that $\tilde{\lambda}\leq\lambda$ , $\tilde{\lambda}(p)=\lambda(p)$ . Hence $\tilde{G}$ also attain its maximum at $p$ near $p$ , and $\tilde{\lambda}$ therefore $\tilde{G}$ is smooth near $p$ . By (6.3) we have

(6.4)

\tilde{\lambda}=D^{2}_{X}u-\Gamma^{k}_{ji}u_{t_{j}}X^{i}X^{k}

where $D$ denotes the usual derivative with respect to real coordinates.

Let $A$ be as before (see (4.2)), then at the point $p$ we get

(6.5)

0\geq\frac{\partial\partial_{J}\tilde{G}\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}=\frac{\partial\partial_{J}\tilde{\lambda}\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}+\frac{\frac{1}{4}\partial\partial_{J}|du|_{g}^{2}\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}.

In local coordinates, the first term is

(6.6)

\begin{split}\frac{\partial\partial_{J}\tilde{\lambda}\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}&=\frac{1}{n}\sum_{p=0}^{n-1}\sum_{i\neq p}\frac{\tilde{\lambda}_{2p\overline{2p}}+\tilde{\lambda}_{2p+1\overline{2p+1}}}{\widetilde{\Omega}_{2i2i+1}}\\ &=\frac{1}{n}\sum_{i=0}^{n-1}\sum_{p\neq i}\frac{\tilde{\lambda}_{2p\overline{2p}}+\tilde{\lambda}_{2p+1\overline{2p+1}}}{\widetilde{\Omega}_{2i2i+1}}.\end{split}

Differentiating (6.4) twice gives

(6.7)

\begin{split}\tilde{\lambda}_{p\overline{p}}&=D^{2}_{X}u_{p\overline{p}}-\Gamma^{k}_{jip\overline{p}}u_{t_{j}}X^{i}X^{k}-\Gamma^{k}_{jip}u_{t_{j}t_{\overline{p}}}X^{i}X^{k}-\Gamma^{k}_{ji\overline{p}}u_{t_{j}t_{p}}X^{i}X^{k}\\ &\geq D^{2}_{X}u_{p\overline{p}}-C_{1}(\tilde{\lambda}+1).\end{split}

Here we used Remark 2.9 and the fact that derivatives of $\Gamma^{k}_{ij}$ depend only on $g$ , and the gradient of $u$ is bounded. In addition

|u_{t_{i}t_{j}}|\leq C_{2}(1+\tilde{\lambda}).

By (5.1) and (3.2) we know that

(6.8)

\frac{1}{C_{3}}\leq\widetilde{\Omega}_{2i2i+1}\leq C_{3}.

Applying (6.7) and (6.8) we can estimate (6.6):

(6.9)

\begin{split}\frac{1}{n}\sum_{i=0}^{n-1}\sum_{p\neq i}\frac{\tilde{\lambda}_{2p\overline{2p}}+\tilde{\lambda}_{2p+1\overline{2p+1}}}{\widetilde{\Omega}_{2i2i+1}}&\geq\frac{1}{n}\sum_{i=0}^{n-1}\sum_{p\neq i}\frac{D^{2}_{X}u_{2p\overline{2p}}+D^{2}_{X}u_{2p+1\overline{2p+1}}}{\widetilde{\Omega}_{2i2i+1}}-C_{1}(\tilde{\lambda}+1)\\ &\geq C_{4}\sum_{p=0}^{2n-1}D^{2}_{X}u_{p\overline{p}}-C_{1}(\tilde{\lambda}+1)\end{split}

To deal with the first term of the right hand side, we use equation (5.7)

\log\text{Pf}(\widetilde{\Omega}_{ij})=f+\log\text{Pf}(\Omega_{ij}).

Differentiating twice in direction $X$ , we get

(6.10)

\frac{1}{2}\sum\widetilde{\Omega}^{ij}D^{2}_{X}\widetilde{\Omega}_{ji}=\frac{1}{2}\sum\widetilde{\Omega}^{ik}D_{X}\widetilde{\Omega}_{kl}\widetilde{\Omega}^{lj}D_{X}\widetilde{\Omega}_{ji}+D^{2}_{X}f+D^{2}_{X}\log\text{Pf}(\Omega_{ij}).

As in previous section,

\begin{split}\sum\widetilde{\Omega}^{ik}D_{X}\widetilde{\Omega}_{kl}\widetilde{\Omega}^{lj}D_{X}\widetilde{\Omega}_{ji}=&\sum\frac{D_{X}\widetilde{\Omega}_{2i+12j}D_{X}\widetilde{\Omega}_{2j+12i}+D_{X}\widetilde{\Omega}_{2i2j+1}D_{X}\widetilde{\Omega}_{2j2i+1}}{\widetilde{\Omega}_{2i2i+1}\widetilde{\Omega}_{2j2j+1}}\\ &-\sum\frac{D_{X}\widetilde{\Omega}_{2i+12j+1}D_{X}\widetilde{\Omega}_{2j2i}+D_{X}\widetilde{\Omega}_{2i2j}D_{X}\widetilde{\Omega}_{2j+12i+1}}{\widetilde{\Omega}_{2i2i+1}\widetilde{\Omega}_{2j2j+1}}.\end{split}

Notice that for $p=0,\dots,2n-1$ ,

\frac{\partial}{\partial t_{p}}\widetilde{\Omega}_{ij}=\frac{\partial}{\partial z^{p}}\widetilde{\Omega}_{ij}+\frac{\partial}{\partial\bar{z}^{p}}\widetilde{\Omega}_{ij},\quad\frac{\partial}{\partial t_{2n+p}}\widetilde{\Omega}_{ij}=-i\big{(}\frac{\partial}{\partial\bar{z}^{p}}\widetilde{\Omega}_{ij}-\frac{\partial}{\partial z^{p}}\widetilde{\Omega}_{ij}\big{)}.

Hence by (2.12), we obtain

\begin{split}&D_{X}\widetilde{\Omega}_{2i2j}=\overline{D_{X}\widetilde{\Omega}_{2i+12j+1}}\,,\quad D_{X}\widetilde{\Omega}_{2i2j+1}=\overline{D_{X}\widetilde{\Omega}_{2j2i+1}}\,,\\ &D_{X}\widetilde{\Omega}_{2i+12j}=\overline{D_{X}\widetilde{\Omega}_{2j+12i}}\,,\quad D_{X}\widetilde{\Omega}_{2i+12j+1}=\overline{D_{X}\widetilde{\Omega}_{2i2j}}\,.\end{split}

Therefore

\sum\widetilde{\Omega}^{ik}D_{X}\widetilde{\Omega}_{kl}\widetilde{\Omega}^{lj}D_{X}\widetilde{\Omega}_{ji}\geq 0.

Combining with (6.10) gives

(6.11)

\sum_{i=0}^{n-1}\frac{D^{2}_{X}\widetilde{\Omega}_{2i2i+1}}{\widetilde{\Omega}_{2i2i+1}}\geq D^{2}_{X}f+D^{2}_{X}\log\text{Pf}(\Omega_{ij}).

Write $J$ in local coordinates as

J=J^{l}_{\bar{k}}d\bar{z}^{k}\otimes\partial_{z^{l}}+J^{\bar{l}}_{k}dz^{k}\otimes\partial_{\bar{z}^{l}}.

Notice that

\widetilde{\Omega}_{ij}=(\Omega_{h})_{ij}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega_{ij}-(-u_{i\overline{k}}J^{\overline{k}}_{j}+u_{j\overline{k}}J^{\overline{k}}_{i})).

Differentiating twice we get

(6.12)

\begin{split}(n-1)D^{2}_{X}\widetilde{\Omega}_{2i2i+1}=&\,(n-1)D^{2}_{X}(\Omega_{h})_{2i2i+1}+\sum_{p=0}^{2n-1}D^{2}_{X}u_{p\overline{p}}+S_{1}(\partial\partial_{J}u)D_{X}^{2}\Omega_{2i2i+1}\\ &-(D^{2}_{X}u_{2i\overline{2i}}+D^{2}_{X}u_{2i+1\overline{2i+1}})+u_{i\overline{k}}D^{2}_{X}J^{\overline{k}}_{j}-u_{j\overline{k}}D^{2}_{X}J^{\overline{k}}_{i}.\end{split}

Here we used Remark 2.9 again, namely at the point $p$ ,

J^{l}_{\bar{k},i}=J^{\bar{l}}_{k,i}=J^{\bar{l}}_{k,\bar{i}}=J^{l}_{\bar{k},\bar{i}}=0.

Combine with (6.11)

(6.13)

\sum_{p=0}^{2n-1}D^{2}_{X}u_{p\overline{p}}\geq-C_{5}(\tilde{\lambda}+1).

Then combining with (6.9) we obtain

(6.14)

\frac{1}{n}\sum_{i=0}^{n-1}\sum_{p\neq i}\frac{\tilde{\lambda}_{2p\overline{2p}}+\tilde{\lambda}_{2p+1\overline{2p+1}}}{\widetilde{\Omega}_{2i2i+1}}\geq-C^{\prime}(\tilde{\lambda}+1).

Now we have the eatimate of (6.6). The second term of (6.5) has been dealt with in $C^{1}$ estimate as in (4.10)

(6.15)

\begin{split}&\frac{\frac{1}{4}\partial\partial_{J}|du|_{g}^{2}\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}\\ =&-\frac{1}{n}\sum_{i=0}^{n-1}\sum_{j=0}^{2n-1}\frac{(\Omega_{h})_{2i2i+1,\overline{j}}u_{j}+(\overline{\Omega}{\mathstrut}_{h})_{2i2i+1,j}u_{\overline{j}}}{\widetilde{\Omega}_{2i2i+1}}+\frac{1}{n}\sum_{j=0}^{2n-1}\frac{u_{j}(e^{f})_{\overline{j}}+u_{\overline{j}}(e^{f})_{j}}{e^{f}}\\ &+\frac{1}{n}\sum_{k=0}^{n-1}\sum_{j=0}^{2n-1}\sum_{i\neq k}\frac{|u_{2kj}|^{2}+|u_{2k+1j}|^{2}}{\widetilde{\Omega}_{2i2i+1}}+\frac{1}{n}\sum_{k=0}^{n-1}\sum_{j=0}^{2n-1}\sum_{i\neq k}\frac{|u_{2k\overline{j}}|^{2}+|u_{2k+1\overline{j}}|^{2}}{\widetilde{\Omega}_{2i2i+1}}.\end{split}

Combining with (6.8) and $C^{1}$ estimate we obtain

(6.16)

\frac{\frac{1}{4}\partial\partial_{J}|du|_{g}^{2}\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}\geq-C_{6}+C_{7}(|u_{ij}|^{2}+|u_{i\bar{j}}|^{2}).

By the definition of $\lambda$

(6.17)

|u_{ij}|^{2}+|u_{i\bar{j}}|^{2}\geq C_{8}\tilde{\lambda}^{2}.

Combining (6.16) and (6.17) we get

(6.18)

\frac{\frac{1}{4}\partial\partial_{J}|du|_{g}^{2}\wedge A\wedge\overline{\Omega}{\mathstrut}^{n}}{\widetilde{\Omega}{\mathstrut}^{n}\wedge\overline{\Omega}{\mathstrut}^{n}}\geq-C_{6}+C\tilde{\lambda}^{2}.

Insert (6.14) and (6.18) into (6.5)

(6.19)

0\geq C\tilde{\lambda}^{2}-C^{\prime}\tilde{\lambda}-C^{\prime\prime}.

This gives upper bound of $\tilde{\lambda}$ , therefore $\lambda$ is bounded above. ∎

7. Proof of the Main Theorem

Once we have the $C^{2}$ estimates, the $C^{2,\alpha}$ -estimates can be derived. In order to prove the main theorem, We consider the following continuity equation $(u_{t},b_{t})$ with $t\in[0,1]$ :

(7.1)		$\displaystyle(\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u_{t})\Omega-\partial\partial_{J}u_{t}))^{n}=e^{tf+(1-t)f_{0}+b_{t}}\Omega^{n},$
(7.2)		$\displaystyle\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u_{t})\Omega-\partial\partial_{J}u_{t})>0,\quad\sup\limits_{M}u_{t}=0.$

where $f_{0}=\log(\Omega_{h}^{n}/\Omega^{n})$ . Consider the set

S=\{t\in[0,1]:(u_{t},b_{t})\in C^{2,\alpha}(M,\mathbb{R})\times\mathbb{R}\text{\ solves the equation\ }\eqref{main1},\eqref{main2}\}

Clearly we have $0\in S$ . The $C^{2,\alpha}$ -estimates implies closedness of $S$ . We would like to show the openness as in [7]. Denote

\widetilde{\Omega}_{u}=\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}u)\Omega-\partial\partial_{J}u).

Consider the operator

\mathcal{M}:\mathcal{A}\ni u\mapsto\frac{\widetilde{\Omega}_{u}^{n}}{\Omega_{u}}\in\mathcal{B},

where

	$\displaystyle\mathcal{A}\coloneqq\{u\in C^{k+2,\alpha}(M):\widetilde{\Omega}_{u}>0,\int_{M}u\Omega^{n}\wedge\overline{\Omega}{\mathstrut}^{n}=0\}$
	$\displaystyle\mathcal{B}\coloneqq\{\tilde{f}\in C^{k,\alpha}(M):\int_{M}\tilde{f}\Omega^{n}\wedge\overline{\Omega}{\mathstrut}^{n}=\int_{M}\Omega^{n}\wedge\overline{\Omega}{\mathstrut}^{n}\}.$

It remains to show that for every $u\in\mathcal{A}$ the differential $d_{u}\mathcal{M}$ is an isomorphism. Indeed for $v\in T_{u}\mathcal{A}$ we have

	$\displaystyle d_{u}\mathcal{M}(v)$	$\displaystyle=\frac{d}{dt}\bigg{\|}_{t=0}\mathcal{M}(u+tv)=\frac{d}{dt}\bigg{\|}_{t=0}\frac{(\Omega_{h}+\frac{1}{n-1}(S_{1}(\partial\partial_{J}(u+tv))\Omega-\partial\partial_{J}(u+tv)))^{n}}{\Omega^{n}}$
		$\displaystyle=\frac{n}{n-1}\frac{(S_{1}(\partial\partial_{J}v)\Omega-\partial\partial_{J}v)\wedge\widetilde{\Omega}_{u}^{n-1}}{\Omega^{n}}=\frac{S_{n-1}(\widetilde{\Omega}_{u})-1}{2(n-1)}\Delta_{I,g}v.$

From general elliptic theory we know that the laplacian is a bijection between the space of functions of zero integral on $M$ . Thus $\mathcal{M}$ is locally invertible and therefore $S$ is open.

Acknowledgements: Fu is supported by NSFC grant No. 12141104. Zhang is supported by NSFC grant No. 11901102.

References

[1] Alesker, S. Solvability of the quaternionic Monge-Ampère equation on compact manifolds with a flat hyperKähler metric. Adv. Math. 241 (2013), 192–219.
[2] Alesker, S., and Shelukhin, E. A uniform estimate for general quaternionic calabi problem (with appendix by daniel barlet). Advances in Mathematics 316 (2017), 1–52.
[3] Alesker, S., and Verbitsky, M. Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry. J. Geom. Anal. 16, 3 (2006), 375–399.
[4] Alesker, S., and Verbitsky, M. Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds. Israel J. Math. 176 (2010), 109–138.
[5] Bedulli, L., Gentili, G., and Vezzoni, L. A parabolic approach to the Calabi-Yau problem in HKT geometry. Math. Z. 302, 2 (2022), 917–933.
[6] Błocki, Z. A gradient estimate in the Calabi-Yau theorem. Math. Ann. 344, 2 (2009), 317–327.
[7] Błocki, Z. The complex Monge-Ampère equation in Kähler geometry. In Pluripotential theory, vol. 2075 of Lecture Notes in Math. Springer, Heidelberg, 2013, pp. 95–141.
[8] Demailly, J.-P. Complex analytic and differential geometry. Citeseer, 1997.
[9] Dinew, S., and Sroka, M. HKT from HK metrics. arXiv preprint arXiv:2105.09344 (2021).
[10] Fu, J., Wang, Z., and Wu, D. Form-type Calabi-Yau equations. Math. Res. Lett. 17, 5 (2010), 887–903.
[11] Fu, J., Wang, Z., and Wu, D. Form-type equations on Kähler manifolds of nonnegative orthogonal bisectional curvature. Calc. Var. Partial Differential Equations 52, 1-2 (2015), 327–344.
[12] Fu, J.-X., and Yau, S.-T. The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Ampère equation. J. Differential Geom. 78, 3 (2008), 369–428.
[13] Gentili, G., and Vezzoni, L. The quaternionic Calabi conjecture on abelian hypercomplex nilmanifolds viewed as tori fibrations. Int. Math. Res. Not. IMRN, 12 (2022), 9499–9528.
[14] Gentili, G., and Zhang, J. Fully non-linear elliptic equations on compact manifolds with a flat hyperkähler metric. J. Geom. Anal. 32, 9 (2022), Paper No. 229, 38.
[15] Lejmi, M., and Weber, P. Cohomologies on hypercomplex manifolds. In Complex and symplectic geometry, vol. 21 of Springer INdAM Ser. Springer, Cham, 2017, pp. 107–121.
[16] Michelsohn, M. L. On the existence of special metrics in complex geometry. Acta Math. 149, 3-4 (1982), 261–295.
[17] Obata, M. Affine transformations in an almost complex manifold with a natural affine connection. J. Math. Soc. Japan 8 (1956), 345–362.
[18] Sroka, M. The $C^{0}$ estimate for the quaternionic Calabi conjecture. Adv. Math. 370 (2020), 107237.
[19] Sroka, M. Sharp uniform bound for the quaternionic Monge-Ampère equation on hyperhermitian manifolds. arXiv e-prints (Nov. 2022), arXiv:2211.00959.
[20] Tosatti, V., and Weinkove, B. The complex Monge-Ampère equation on compact Hermitian manifolds. J. Amer. Math. Soc. 23, 4 (2010), 1187–1195.
[21] Tosatti, V., and Weinkove, B. Estimates for the complex Monge-Ampère equation on Hermitian and balanced manifolds. Asian J. Math. 14, 1 (2010), 19–40.
[22] Tosatti, V., and Weinkove, B. The Monge-Ampère equation for $(n-1)$ -plurisubharmonic functions on a compact Kähler manifold. J. Amer. Math. Soc. 30, 2 (2017), 311–346.
[23] Tosatti, V., and Weinkove, B. Hermitian metrics, $(n-1,n-1)$ forms and Monge-Ampère equations. J. Reine Angew. Math. 755 (2019), 67–101.
[24] Verbitsky, M. HyperKähler manifolds with torsion, supersymmetry and Hodge theory. Asian Journal of Mathematics 6 (01 2002), 679–712.
[25] Verbitsky, M. Balanced HKT metrics and strong HKT metrics on hypercomplex manifolds. Math. Res. Lett. 16, 4 (2009), 735–752.
[26] Verbitsky, M. Positive forms on hyperkähler manifolds. Osaka J. Math. 47, 2 (2010), 353–384.
[27] Yau, S. T. On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I. Comm. Pure Appl. Math. 31, 3 (1978), 339–411.

The Monge-Ampère equation for (n−1)(n-1)-quaternionic PSH functions on a hyperKähler manifold

Abstract.

1. Introduction

Theorem 1.1.

Corollary 1.2.

2. Preliminaries and notation

Lemma 2.1 ([9], Lemma 2.12).

Definition 2.2.

Lemma 2.3 ([18], Lemma 3).

Definition 2.4.

Definition 2.5.

Definition 2.6.

Remark 2.7.

Lemma 2.8.

Proof.

Remark 2.9.

3. C0C^{0} Estimate

Theorem 3.1.

Lemma 3.2.

Proof.

Lemma 3.3.

Proof.

Lemma 3.4.

Proof.

Proof of Theorem 3.1.

4. C1C^{1} Estimate

Theorem 4.1.

Proof.

5. Bound on ∂∂Ju\partial\partial_{J}u

Theorem 5.1.

Proof.

6. C2C^{2} Estimate

Theorem 6.1.

Proof.

7. Proof of the Main Theorem

References

The Monge-Ampère equation for $(n-1)$ -quaternionic PSH functions on a hyperKähler manifold

3. $C^{0}$ Estimate

4. $C^{1}$ Estimate

5. Bound on $\partial\partial_{J}u$

6. $C^{2}$ Estimate