Form-type equations on Kähler manifolds of nonnegative orthogonal bisectional curvature

Abstract

In this paper we prove the existence and uniqueness of the form-type equation on Kähler manifolds of nonnegative orthogonal bisectional curvature.

1 Introduction

In the previous paper [2], we introduced the form-type Calabi–Yau equation on a compact complex $n$-dimensional manifold with a balanced metric and with a non-vanishing holomorphic $n$-form $\Omega$. A balanced metric $\omega$ on $X$ is a hermitian metric such that $d\omega ^{n-1}=0$. Given a balanced metric $\omega _0$ on $X$, let us denote by ${\mathcal P}(\omega _0)$ the set of all smooth real $(n-2,n-2)$-forms $\psi$ such that $\omega _0^{n-1}+\frac{\sqrt{-1}}{2}\partial \bar{\partial }\psi >0$ on $X$. Then, for each $\varphi \in {\mathcal P}(\omega _0)$, there exists a balanced metric, which we denote by $\omega _{\varphi }$, such that $\omega _{\varphi }^{n-1}=\omega _0^{n-1}+\frac{\sqrt{-1}}{2}\partial \bar{\partial }\varphi$. We say that such a metric $\omega _\varphi$ is in the balanced class of $\omega _0$. Our aim is to find a balanced metric $\omega _\varphi$ in the balanced class of $\omega _0$ such that

$$\begin{aligned} \Vert \Omega \Vert _{\omega _\varphi }=\mathrm{a\,constant}\ C_0>0. \end{aligned}$$

(1.1)

The geometric meaning of such a metric is that its Ricci curvatures of the hermitian connection and the spin connection are zero. On the other hand, the direct non-Kähler analogue of the Calabi conjecture has recently been solved by Tosatti–Weinkove [9] (see also [5], and the references in [5, 9]). In general their solutions provide hermitian Ricci-flat metrics which are not balanced.

As in the Kähler case, Eq. (1.1) can be reformulated in the following form

$$\begin{aligned} \frac{\omega _\varphi ^n}{\omega _0^n} = e^f \frac{\int _X \omega _\varphi ^n }{\int _X \omega _0^n}, \end{aligned}$$

(1.2)

where $f \in C^{\infty }(X)$ is given and satisfies the compatibility condition:

$$\begin{aligned} \int _X e^f \omega _0^n = \int _X \omega _0^n. \end{aligned}$$

(1.3)

We would like to find a solution $\varphi \in {\mathcal {P}}(\omega _0)$. The Eq. (1.2) is called a form-type Calabi–Yau equation, a reminiscent of the classic function type Calabi–Yau equation. We note here that when $n=2$, the form type equation is reduced automatically to the classic function type equation and the balanced metric is a Kähler metric. Hence in this case the Eq. (1.2) is the classic Calabi–Yau equation and has been solved by Yau in [10]. Therefore, in the following we assume $n\ge 3$.

We have constructed solutions for (1.1) when $X$ is a complex torus [2]. A natural approach to solve (1.2) is to use the continuity method. The openness and uniqueness were discussed in the previous work [2]. We do not know whether there is a geometric obstruction for solving (1.2) in general.

Equation (1.2) is still meaningful on a compact complex manifold with a balanced metric, whose canonical bundle is not holomorphically trivial. Geometrically, solving (1.2) allows us to solve the problem of prescribed volume form on $X$, in the balanced class of each balanced metric on $X$. Namely, given any positive $(n,n)$-form $W$ on $X$ and a balanced metric $\omega _0$, we let

$$\begin{aligned} e^f = \left( \frac{W}{\omega _0^n}\right) \frac{\int _X \omega _0^n}{\int _X W}; \end{aligned}$$

then by solving (1.2) we are able to find a metric $\omega _\varphi$ in the balanced class of $\omega _0$ such that $\omega _\varphi ^n$ is equal to $W$, up to a constant rescaling.

It seems to us very hard to understand Eq. (1.2) in general. In this paper, we want to give the mechanics of looking for all solutions within the balanced class of a given balanced metric. The idea is, in some sense, to transfer the form-type Calabi–Yau equation to a function type equation.

So in the following we let $(X, \eta )$ be an $n$-dimensional Kähler manifold, $n \ge 3$, and $\omega _0$ be a balanced metric on $X$. We let on $X$

$$\begin{aligned} {\mathcal {P}}_{\eta }(\omega _0) = \left\{ v \in C^{\infty }(X)\mid \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }v \wedge \eta ^{n-2} >0 \right\} \!. \end{aligned}$$

For each $u \in {\mathcal {P}}_{\eta }(\omega _0)$, we denote by $\omega _u$ the unique positive $(1,1)$-form on $X$ such that

$$\begin{aligned} \omega _u^{n - 1} = \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u \wedge \eta ^{n-2} \quad \mathrm{on}\,X. \end{aligned}$$

Then we consider the equation

$$\begin{aligned} \frac{\omega _u^n}{\omega _0^n} = e^f \frac{\int _X \omega _u^n }{\int _X \omega _0^n}, \end{aligned}$$

(1.4)

where $f \in C^{\infty }(X)$ is given and satisfies the compatibility condition (1.3).

In this paper, we are able to solve (1.4), under the assumption that the Kähler metric $\eta$ has nonnegative orthogonal bisectional curvature; that is, for any orthonormal tangent frame $\{ e_1, \ldots , e_n\}$ at any $x\in M$, the curvature tensor of $\eta$ satisfies that

$$\begin{aligned} R_{i\bar{i}j\bar{j}} \equiv R (e_i, \bar{e}_i, e_j, \bar{e}_j) \ge 0, \quad \mathrm{for\,all}\,1 \le i,j \le n\,\mathrm{and}\,i \ne j. \end{aligned}$$

(1.5)

We remark that nonnegativity of the orthogonal bisectional curvature is weaker than nonnegativity of the bisectional curvature. In fact, the former condition are satisfied by not only complex projective spaces and the Hermitian symmetric spaces, but also some compact Kähler manifolds of dimension $\ge 2$ whose holomorphic sectional curvature is strictly negative somewhere. We refer the reader to the recent work Gu–Zhang [4] for the study of nonnegative orthogonal bisectional curvature, which generalizes the earlier work of Mok [7] and Siu–Yau [8].

Our main result is as follows:

Theorem 1

Let $(X,\eta )$ be a compact Kähler manifold of nonnegative orthogonal bisectional curvature, and $\omega _0$ be a balanced metric on $X$. Then, for any smooth function $f$ on $X$ satisfying (1.3), Eq. (1.4) admits a solution $u \in {\mathcal {P}}_{\eta }(\omega _0)$, which is unique up to a constant.

Subsequently, Eq. (1.2) has a solution $\varphi =u\eta ^{n-2} \in {\mathcal {P}}(\omega _0)$. Now we explain how to use Theorem 1 to find all solutions of (1.2) in the balanced class of $\omega _0$ on a compact Kähler manifold $(X,\eta )$ of nonnegative orthogonal bisectional curvature. Let $\omega _\psi$, for $\psi \in {\mathcal {P}}(\omega _0)$, be a balanced metric in the balanced class of $\omega _0$. We then let

$$\begin{aligned} {\mathcal {P}}_{\eta }(\omega _\psi ) = \left\{ v \in C^{\infty }(X)\mid \omega _\psi ^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }v \wedge \eta ^{n-2} >0 \right\} \!. \end{aligned}$$

For each $v \in {\mathcal {P}}_{\eta }(\omega _\psi )$, we denote by $\omega _{\psi ,v}$ the unique positive (1,1)-form on $X$ such that

$$\begin{aligned} \omega _{\psi ,v}^{n - 1} = \omega _\psi ^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }v \wedge \eta ^{n-2}. \end{aligned}$$

Such a $(1,1)$-form $\omega _{\psi ,v}$ is still in the balanced class of $\omega _0$. Then we consider the equation

$$\begin{aligned} \frac{\omega _{\psi ,u}^n}{\omega _\psi ^n} = e^{f_\psi } \frac{\int _X \omega _{\psi ,u}^n }{\int _X \omega _\psi ^n}, \end{aligned}$$

(1.6)

where $f_\psi \in C^{\infty }(X)$ is given by

$$\begin{aligned} e^{f_\psi }=e^f\frac{\omega _0^n}{\omega _\psi ^n}\frac{\int _X\omega _\psi ^n}{\int _X\omega _0^n} \end{aligned}$$

and satisfies the compatibility condition.

Replacing $\omega _0$ with $\omega _\psi$ and $f$ with $f_\psi$ in Theorem 1, we show that (1.6) admits a solution, denoted by $u_{\psi }$, which is unique up to a constant. It then follows that $\varphi =\psi +u_\psi \eta ^{n-2} \in {\mathcal {P}}(\omega _0)$ is a solution to (1.2). Hence, when we vary $\psi \in {\mathcal P}(\omega _0)$, we obtain all solutions $\varphi =\psi +u_\psi \eta ^{n-2}$ (which are infinitely many) to Eq. (1.2) in the balanced class of $\omega _0$ on $X$ of nonnegative orthogonal bisectional curvature. In particular, the form-type equation on a complex torus is completely settled in this way.

Corollary 2

Let $(X,\eta )$ be a compact Kähler manifold of nonnegative orthogonal bisectional curvature, and $\omega _0$ be a balanced metric on $X$. Let $f$ be a smooth function on $X$ satisfying (1.3). Then for any $\psi \in {\mathcal P}(\omega _0)$, Eq. (1.2) admits a solution $\varphi =\psi +u_\psi \eta ^{n-2}$. Here $u_\psi$ is a solution to (1.6) which is unique up to a constant.

Thus, the idea used in this paper, which is to transfer from the form-type Calabi–Yau equation to a function-type equation, may be useful. Later we will establish the Theorem 1 on any compact Kähler manifold. We need to overcome some difficulties of estimates.

We employ the continuity method to prove Theorem 1. In Sect. 2, we establish an a priori $C^2$ estimate for the solution $u$. This is the place where we need the curvature condition. The $C^2$ estimate enables us to obtain a general a priori $C^0$ estimate, by combining the maximum principle and the weak Harnack inequality. This is the content of Sect. 3. We then adapt the Evans–Krylov theory to our form-type equation, and obtain in Sect. 4 the Hölder estimates for second derivatives. The openness is covered by Theorem 3 in our previous paper [2]. For readers’ convenience, we briefly indicate the argument in the last section, Sect. 5. The uniqueness is also proved in Sect. 5.

2 $C^2$ estimates for form-type equations

In this section, we would like to establish the following estimate:

Lemma 3

Given $F \in C^2(X)$, let $u \in C^4(X)$ satisfy that

$$\begin{aligned} \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u \wedge \eta ^{n-2} > 0 \quad \mathrm{on}\,X, \end{aligned}$$

and that

$$\begin{aligned} \det \left[ \omega _0^{n-1}+(\sqrt{-1}/2) \partial \bar{\partial }u\wedge \eta ^{n-2}\right] = e^{F} \det \omega _0^{n-1}. \end{aligned}$$

(2.1)

Assume that $\eta$ has nonnegative orthogonal bisectional curvature. Then, we have

$$\begin{aligned} \Delta _{\eta } u \le C + C( u - \inf _X u) \qquad \mathrm{on}\,X, \end{aligned}$$

(2.2)

and

$$\begin{aligned} \sup _X |\omega _0^{n-1} + \partial \bar{\partial } u \wedge \eta ^{n-2}|_{\eta } \le C + \left( \sup _X u - \inf _X u \right) . \end{aligned}$$

Here $\Delta _{\eta } v =\sum \eta ^{i\bar{j}} v_{i\bar{j}}$ denotes the Laplacian of a function $v$ with respect to $\eta$, and $C>0$ is a constant depending only on $\inf _X (\Delta _{\eta }F)$, $\sup _X F$, $\eta$, $n$, and $\omega _0$.

Here are some conventions: For an $(n-1,n-1)$-form $\Theta$, we denote

$$\begin{aligned} \Theta&= \left( \frac{\sqrt{-1}}{2} \right) ^{n-1} (n-1)! \\&\quad \times \sum _{p,q}s(p,q)\Theta _{p\bar{q}}dz^1\wedge d\bar{z}^1 \cdots \wedge \widehat{dz^{p}} \wedge d \bar{z}^p \wedge \cdots \wedge d\bar{z}^q \wedge \widehat{d\bar{z}^{q}}\wedge \cdots \wedge d z^n \wedge d\bar{z}^n, \end{aligned}$$

in which

$$\begin{aligned} s(p,q) = {\left\{ \begin{array}{ll} - 1, &{} \mathrm{if}\, p > q; \\ 1, &{} \mathrm{if}\,p \le q. \end{array}\right. } \end{aligned}$$

(2.3)

Here we introduce the sign function $s$ so that,

$$\begin{aligned}&d z^p \wedge d\bar{z}^q \wedge s(p,q) dz^1\wedge d\bar{z}^1 \cdots \wedge \widehat{dz^p} \wedge d \bar{z}^p \wedge \cdots \wedge d\bar{z}^q \wedge \widehat{d\bar{z}^q}\wedge \cdots \wedge d z^n \wedge d\bar{z}^n \\&\quad = dz^1 \wedge d\bar{z}^1 \wedge \cdots \wedge d z^n \wedge d\bar{z}^n, \quad \mathrm{for\,all} \quad 1 \le p, q \le n. \end{aligned}$$

We denote

$$\begin{aligned} \det \Theta = \det (\Theta _{p\bar{q}}). \end{aligned}$$

If the matrix $(\Theta _{p\bar{q}})$ is invertible, we denote by $(\Theta ^{p\bar{q}})$ the transposed inverse of $(\Theta _{p\bar{q}})$, i.e.,

$$\begin{aligned} \sum _l \Theta _{i\bar{l}} \Theta ^{j\bar{l}} = \delta _{ij}. \end{aligned}$$

Note that, for a positive $(1,1)$-form $\omega$ given by

$$\begin{aligned} \omega = \frac{\sqrt{-1}}{2}\sum _{i,j=1}^n g_{i\bar{j}} dz_i \wedge d\bar{z}_j, \end{aligned}$$

we have

$$\begin{aligned} \omega ^n = \left( \frac{\sqrt{-1}}{2} \right) ^{n} n! \det (g_{i\bar{j}}) dz^1\wedge d\bar{z}_1 \wedge \cdots \wedge dz^n \wedge d\bar{z}^n, \end{aligned}$$

and by our convention,

$$\begin{aligned} (\omega ^{n-1})_{i\bar{j}} = \det (g_{i\bar{j}}) g^{i\bar{j}}. \end{aligned}$$

It follows that

$$\begin{aligned} \det (\omega ^{n-1}) = \det (g_{i\bar{j}})^{n-1}, \end{aligned}$$

(2.4)

and

$$\begin{aligned} (\omega ^{n-1})^{i\bar{j}} = \frac{g_{i\bar{j}}}{\det (g_{i\bar{j}})}. \end{aligned}$$

In the following, the subscripts such as “$,p$” stand for the ordinary local derivatives; for example,

$$\begin{aligned} \eta _{i\bar{j},k} = \frac{\partial \eta _{i\bar{j}}}{\partial z^k}, \quad \eta _{i\bar{j},l\bar{m}} = \frac{\partial ^2 \eta _{i\bar{j}}}{\partial z^l \partial \bar{z}^m}. \end{aligned}$$

(2.5)

For a function $h$ we can omit the comma: $h_l = h_{,l}$, $h_{l\bar{m}} = h_{,l\bar{m}}$, etc. Unless otherwise indicated, all the summations below range from $1$ to $n$. We remark that, under the convention, Eq. (1.4) can be rewritten as

$$\begin{aligned} \frac{\det [\omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u \wedge \eta ^{n-2}]}{\det \omega _0^{n-1}} = e^{(n-1)f} \left( \frac{\int _X \omega _u^n }{\int _X \omega _0^n} \right) ^{n-1}, \end{aligned}$$

which is convenient for deriving the estimates.

Proof of Lemma 3

Let

$$\begin{aligned} \Psi _u=\Psi +(\sqrt{-1}/2) \partial \bar{\partial }u\wedge \eta ^{n-2}, \quad \text {where} \quad \Psi =\omega _0^{n-1}. \end{aligned}$$

Let

$$\begin{aligned} \phi&= \frac{\sum _{i,j} \eta _{i\bar{j}} (\Psi _u)_{i\bar{j}}}{\det \eta } - A u, \end{aligned}$$

where $A >0$ is a large constant to be determined. Using wedge products, the function $\phi$ can also be written as

$$\begin{aligned} \phi&= \frac{n \eta \wedge \Psi _u}{\eta ^n} - Au \nonumber \\&= (h + \Delta _{\eta } u) - A u, \quad \mathrm{where}\,h = \frac{n \eta \wedge \omega _0^{n-1}}{\eta ^n}. \end{aligned}$$

(2.6)

Consider the operator

$$\begin{aligned} L\phi =(n-1)\sum _{k,l} \Psi _{u}^{k\bar{l}}\left( \frac{\sqrt{-1}}{2}\partial \bar{\partial }\phi \wedge \eta ^{n-2} \right) _{k\bar{l}}. \end{aligned}$$

Suppose that $\phi$ attains its maximum at some point $P$ in $X$. We choose a normal coordinate system such that at $P$, $\eta _{i\bar{j}}=\delta _{ij}$ and $d\eta _{i\bar{j}}=0$. Then, we rotate the axes so that at $P$ we have $(\Psi _u)_{p\bar{q}}=\delta _{pq}(\Psi _u)_{p\bar{p}}$. Thus, for any smooth function $v$ on $X$, we have at $P$ that

$$\begin{aligned} (n-1)\left( \frac{\sqrt{-1}}{2}\partial \bar{\partial } v \wedge \eta ^{n-2}\right) _{i\bar{j}} = \delta _{ij}\sum _{p \ne i} v_{p\bar{p}} + (1 - \delta _{ij})v_{j\bar{i}}. \end{aligned}$$

(2.7)

By (2.7) we obtain that

$$\begin{aligned} (\Psi _u)_{i\bar{i}}&= \Psi _{i\bar{i}} + \frac{1}{n-1} \sum _{q \ne i} u_{q\bar{q}}, \end{aligned}$$

(2.8)

$$\begin{aligned} (\Psi _u)_{i\bar{j}}&= \Psi _{i\bar{j}} + \frac{u_{j\bar{i}}}{n - 1} = 0, \quad \mathrm{for\,all}\,i \ne j. \end{aligned}$$

(2.9)

It follows that

$$\begin{aligned} \sum _{i=1}^n (\Psi _u)_{i\bar{i}} = \sum _{i=1}^n \Psi _{i\bar{i}} + \sum _{i=1}^n u_{i\bar{i}} = h + \Delta _{\eta } u. \end{aligned}$$

(2.10)

Furthermore, we have

$$\begin{aligned} (\Psi _u)_{i\bar{j},p} = \Psi _{i\bar{j},p} + \frac{\delta _{ij}}{n-1} \sum _{q \ne i} u_{q\bar{q}p} + \frac{1 - \delta _{ij}}{n - 1} u_{j\bar{i}p}, \end{aligned}$$

(2.11)

and

$$\begin{aligned} (\Psi _u)_{i\bar{i},p\bar{p}}&= \Psi _{i\bar{i},p\bar{p}} + \frac{1}{n-1}\sum _{k \ne i} u_{k\bar{k}p\bar{p}} + \frac{1}{n - 1}\sum _{k \ne i} u_{k\bar{k}}\left( \,\sum _{j \ne k, j \ne i} \eta _{j\bar{j},p\bar{p}} \right) \\&\quad - \frac{1}{n - 1}\sum _{a \ne i, b \ne i, a \ne b} u_{a\bar{b}} \eta _{b\bar{a},p\bar{p}}. \end{aligned}$$

Note that under the normal coordinate system, the curvature $(R_{i\bar{j}k\bar{l}})$ of $\eta$ reads

$$\begin{aligned} R_{i\bar{j}k\bar{l}} = - \eta _{i\bar{j},k\bar{l}} + \sum _{a,b} \eta ^{a\bar{b}} \eta _{i\bar{b},k} \eta _{a\bar{j},\bar{l}} = - \eta _{i\bar{j},k\bar{l}}, \quad \mathrm{at} \quad P. \end{aligned}$$

This together with (2.9) imply that

$$\begin{aligned} (\Psi _u)_{i\bar{i},p\bar{p}}&= \Psi _{i\bar{i},p\bar{p}} + \frac{1}{n-1}\sum _{k \ne i} u_{k\bar{k}p\bar{p}} - \frac{1}{n - 1}\sum _{k \ne i} u_{k\bar{k}}\left( \,\sum _{j \ne k, j \ne i} R_{j\bar{j}p\bar{p}} \right) \nonumber \\&\quad - \sum _{a \ne i, b \ne i, a \ne b} \Psi _{a\bar{b}} R_{a\bar{b}p\bar{p}}. \end{aligned}$$

(2.12)

We compute at $P$ that

$$\begin{aligned} L\phi =&(n-1)\sum _l (\Psi _u)^{l\bar{l}} \left( \frac{\sqrt{-1}}{2}\partial \bar{\partial } \phi \wedge \eta ^{n-2}\right) _{l\bar{l}} = \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}} \phi _{p\bar{p}}. \end{aligned}$$

Note that

$$\begin{aligned} 0 = \phi _p (P) = h_p + (\Delta _{\eta } u)_p - A u_p. \end{aligned}$$

(2.13)

Differentiating once more to obtain that

$$\begin{aligned} 0 \ge \phi _{p\bar{p}} (P) = h_{p\bar{p}} + (\Delta _{\eta } u)_{p\bar{p}} - A u_{p\bar{p}}. \end{aligned}$$

It follows that

$$\begin{aligned} 0&\ge L \phi = \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}} \phi _{p\bar{p}}\nonumber \\&= \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}}[h_{p\bar{p}} + (\Delta _{\eta } u)_{p\bar{p}}] - A \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}} u_{p\bar{p}}. \end{aligned}$$

(2.14)

Notice that

$$\begin{aligned}&\sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}}[h_{p\bar{p}} + (\Delta _{\eta } u)_{p\bar{p}}]\nonumber \\&\quad = \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}} h_{p\bar{p}} + \sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}} u_{a\bar{a}p\bar{p}} + \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}} \sum _{a,b} \eta ^{a\bar{b}}_{,p\bar{p}} u_{a\bar{b}}\nonumber \\&\quad = \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}} h_{p\bar{p}} + \sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}} u_{a\bar{a}p\bar{p}} + \sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}} R_{a\bar{a}p\bar{p}} u_{a\bar{a}}\nonumber \\&\qquad - (n-1) \sum _l \sum _{p \ne l}\sum _{a \ne b} (\Psi _u)^{l\bar{l}}R_{b\bar{a}p\bar{p}} \Psi _{b\bar{a}}, \qquad \left( \mathrm{by}\,(2.9)\right) \!. \end{aligned}$$

(2.15)

Here the fourth derivative term can be handled by the Eq. (2.1): We rewrite (2.1) as

$$\begin{aligned} \log \det \Psi _u = F + \log \det \Psi . \end{aligned}$$

Differentiating this in the direction of $\partial /\partial z^a$ yields

$$\begin{aligned} \sum _{k,l} (\Psi _u)^{k\bar{l}}(\Psi _u)_{k\bar{l},a} = (F + \log \det \Psi )_{a}. \end{aligned}$$

Then,

$$\begin{aligned} \sum _{k,l}(\Psi _u)^{k\bar{l}}(\Psi _u)_{k\bar{l},a\bar{b}}=(F + \log \det \Psi )_{a\bar{b}} +\sum _{k,l,p,q}(\Psi _u)^{k\bar{q}}(\Psi _u)^{p\bar{l}}(\Psi _u)_{k\bar{l},a} (\Psi _u)_{p\bar{q},\bar{b}}. \end{aligned}$$

Contracting this with $(\eta ^{a\bar{b}})$ and applying the normal coordinates yield that

$$\begin{aligned} \sum _{l,a}(\Psi _u)^{l\bar{l}}(\Psi _u)_{l\bar{l},a\bar{a}}=\sum _a (F + \log \det \Psi )_{a\bar{a}} +\sum _{k,l,a}\frac{\big |(\Psi _u)_{k\bar{l},a}\big |^2}{(\Psi _u)_{l\bar{l}}(\Psi _u)_{k\bar{k}}}. \end{aligned}$$

This together with (2.12) imply that

$$\begin{aligned}&\sum _{l,a} (\Psi _u)^{l\bar{l}}\Psi _{l\bar{l},a\bar{a}} + \frac{1}{n-1}\sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}} u_{p\bar{p}a\bar{a}} \\&\quad = \sum _{k,l,a}\frac{\big |(\Psi _u)_{k\bar{l},a}\big |^2}{(\Psi _u)_{l\bar{l}}(\Psi _u)_{k\bar{k}}} + \frac{1}{n-1} \sum _{l,a} (\Psi _u)^{l\bar{l}}\sum _{p \ne l} u_{p\bar{p}}\left( \sum _{m \ne p, m \ne l}R_{m\bar{m}a\bar{a}} \right) \\&\qquad + \sum _{l,a} (\Psi _u)^{l\bar{l}} \left( \sum _{p \ne l, q \ne l, p \ne q} \Psi _{p\bar{q}} R_{p\bar{q}a\bar{a}} \right) + \Delta _{\eta } F + \Delta _\eta (\log \det \Psi ). \end{aligned}$$

Combining this with (2.15) yields

$$\begin{aligned}&\sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}}(h_{p\bar{p}} + (\Delta _{\eta } u)_{p\bar{p}})\nonumber \\&\quad = \sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}}R_{a\bar{a}p\bar{p}} u_{a\bar{a}} + \sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}}u_{p\bar{p}}\left( \sum _{m \ne p, m \ne l} R_{a\bar{a}m\bar{m}} \right) \nonumber \\&\qquad + (n-1)\sum _{k,l,a}\frac{\big |(\Psi _u)_{k\bar{l},a}\big |^2}{(\Psi _u)_{l\bar{l}}(\Psi _u)_{k\bar{k}}} +(n-1) \Delta _{\eta } F + (n-1)\Delta _{\eta }(\log \det \Psi )\nonumber \\&\qquad + \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}} h_{p\bar{p}} - (n - 1) \sum _{l, a} (\Psi _u)^{l\bar{l}} \Psi _{l\bar{l},a\bar{a}}\nonumber \\&\qquad + (n-1)\sum _{l,a}(\Psi _u)^{l\bar{l}} \left( \sum _{p \ne l, q \ne l, p \ne q}\Psi _{p\bar{q}} R_{p\bar{q}a\bar{a}} \right) \nonumber \\&\qquad - (n-1)\sum _l \sum _{p \ne l} \sum _{a \ne b} (\Psi _u)^{l\bar{l}} R_{a\bar{b}p\bar{p}} \Psi _{a\bar{b}}. \end{aligned}$$

(2.16)

The first two terms on the right hand side of the above inequality can be handled as follows.

$$\begin{aligned}&\sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}}R_{a\bar{a}p\bar{p}} u_{a\bar{a}} + \sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}}u_{p\bar{p}}\left( \sum _{m \ne p, m \ne l} R_{a\bar{a}m\bar{m}} \right) \nonumber \\&\quad = \sum _{l,a} (\Psi _u)^{l\bar{l}} R_{l\bar{l}a\bar{a}} u_{l\bar{l}} - \sum _{l,a} (\Psi _u)^{l\bar{l}} R_{a\bar{a}l\bar{l}} u_{a\bar{a}} + \sum _{l,p} \sum _{a \ne l} (\Psi _u)^{l\bar{l}} u_{a\bar{a}} R_{a\bar{a}p\bar{p}}\nonumber \\&\qquad +\, \sum _{l,a} \sum _{p \ne l} (\Psi _u)^{l\bar{l}}u_{p\bar{p}}\left( \sum _{m \ne p, m \ne l} R_{a\bar{a}m\bar{m}} \right) \nonumber \\&\quad = \frac{1}{2} \sum _{l,a} (\Psi _u)^{l\bar{l}} R_{l\bar{l}a\bar{a}} (u_{l\bar{l}} - u_{a\bar{a}}) + \frac{1}{2} \sum _{l,a} (\Psi _u)^{a\bar{a}} R_{l\bar{l}a\bar{a}} (u_{a\bar{a}} - u_{l\bar{l}})\nonumber \\&\qquad +\, (n-1)\sum _l \left( \sum _{m \ne l} R_{m\bar{m}}\right) (\Psi _u)^{l\bar{l}} \left[ (\Psi _u)_{l\bar{l}} - \Psi _{l\bar{l}}\right] \quad \left( \mathrm{by}\,(2.8)\right) \nonumber \\&\quad = \frac{1}{2}\sum _{l,a} R_{l\bar{l}a\bar{a}} \frac{(u_{l\bar{l}} - u_{a\bar{a}})[(\Psi _u)_{a\bar{a}} - (\Psi _u)_{l\bar{l}}]}{(\Psi _u)_{l\bar{l}} (\Psi _u)_{a\bar{a}}}\nonumber \\&\qquad +\, (n - 1)^2 \sum _l R_{l\bar{l}} - (n -1) \sum _l (\Psi _u)^{l\bar{l}} \Psi _{l\bar{l}} \left( \sum _{m \ne l} R_{m\bar{m}} \right) . \end{aligned}$$

(2.17)

Apply (2.8) to estimate the first term of last equality

$$\begin{aligned}&\frac{1}{2}\sum _{l,a} R_{l\bar{l}a\bar{a}}\frac{(u_{l\bar{l}} - u_{a\bar{a}})[(\Psi _u)_{a\bar{a}} - (\Psi _u)_{l\bar{l}}]}{(\Psi _u)_{l\bar{l}} (\Psi _u)_{a\bar{a}}}\nonumber \\&\quad = \frac{n - 1}{2}\sum _{l,a} R_{l\bar{l}a\bar{a}}\frac{[(\Psi _u)_{a\bar{a}} - (\Psi _u)_{l\bar{l}}]^2}{(\Psi _u)_{l\bar{l}} (\Psi _u)_{a\bar{a}}}\nonumber \\&\qquad + \frac{n - 1}{2} \sum _{l,a}R_{l\bar{l}a\bar{a}}\frac{(\Psi _{l\bar{l}} - \Psi _{a\bar{a}})[(\Psi _u)_{a\bar{a}} - (\Psi _u)_{l\bar{l}}]}{(\Psi _u)_{l\bar{l}} (\Psi _u)_{a\bar{a}}}\nonumber \\&\quad \ge (n - 1) \sum _{l,a}R_{l\bar{l}a\bar{a}}\frac{\Psi _{l\bar{l}} - \Psi _{a\bar{a}}}{(\Psi _u)_{l\bar{l}}}, \qquad \mathrm{by\,curvature\,assumption } \,\,(1.5). \end{aligned}$$

(2.18)

Combining (2.16) with (2.17) and then with (2.18), we obtain

$$\begin{aligned}&\sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}}(h_{p\bar{p}} + (\Delta _{\eta } u)_{p\bar{p}})\nonumber \\&\quad \ge - C_1(n-1) \sum _l (\Psi _u)^{l\bar{l}} - (n-1)^2 C_1 + (n-1)\inf \Delta _{\eta } F. \end{aligned}$$

(2.19)

Here and throughout this section, we denote by $C_1>0$ a generic constant depending only on $\Psi$ and the curvature of $\eta$.

Substituting (2.19) into (2.14) yields

$$\begin{aligned} 0 \ge L \phi&\ge - A \sum _l \sum _{p \ne l} (\Psi _u)^{l\bar{l}} u_{p\bar{p}} - C_1(n - 1) \sum _l (\Psi _u)^{l\bar{l}} \\&\quad - (n-1)^2 C_1 + (n-1)\inf \Delta _{\eta } F \\&= - n (n-1) A + (n-1) A \sum _l (\Psi _u)^{l\bar{l}} \Psi _{l\bar{l}} - C_1 (n-1) \sum _l (\Psi _u)^{l\bar{l}} \\&\quad - (n-1)^2 C_1 + (n-1) \inf \Delta _{\eta } F. \end{aligned}$$

Now we choose $A>0$ sufficiently large so that

$$\begin{aligned} A \inf _X (\min _l \Psi _{l\bar{l}}) \ge 2 C_1. \end{aligned}$$

It follows that

$$\begin{aligned}&\frac{nA}{C_1}+(n - 1) - \frac{\inf \Delta _{\eta } F}{C_1} \ge \sum _{l=1}^n (\Psi _u)^{l\bar{l}} \\&\quad \ge \left[ \sum _{i=1}^n(\Psi _u)_{i\bar{i}}\right] ^{\frac{1}{n-1}}\left\{ \det \left[ (\Psi _u)_{i\bar{j}}\right] \right\} ^{\frac{-1}{n-1}} \\&\quad = \left[ \sum _{i=1}^n (\Psi _u)_{i\bar{i}}\right] ^{\frac{1}{n-1}}e^{\frac{- F}{n-1}} (\det \Psi )^{\frac{-1}{n-1}}. \end{aligned}$$

Hence,

$$\begin{aligned} h + \Delta _{\eta } u = \sum _{i=1}^n (\Psi _u)_{i\bar{i}} \le C_2 \quad \mathrm{at}\,P. \end{aligned}$$

Here and throughout this section, we denote by $C_2$ a generic positive constant depending only on $n$, $\Psi$, $\eta$, $\sup \Delta _{\eta } F$, and $\sup F$. Therefore, at any point in $X$,

$$\begin{aligned} (h + \Delta _{\eta } u) \le (h + \Delta _{\eta } u)(P) + A u - A u(P) \le C_2 + C_2 \left( u - \inf _X u\right) \!. \end{aligned}$$

Since $[(\Psi _u)_{i\bar{j}}]$ is positive definite everywhere, we have

$$\begin{aligned} |(\Psi _u)_{i\bar{j}}| \le C_2 + C_2 (u - \inf _X u), \quad for all 1 \le i, j \le n . \end{aligned}$$

This completes the proof. $\square$

Lemma 3 enables us to establish the $C^2$ estimate for Eq. (1.4):

Corollary 4

For any $f \in C^{\infty }(X)$, let $u \in C^{\infty }(X)$ be a solution of

$$\begin{aligned} \frac{\det (\omega _u^{n-1})}{\det (\omega _0^{n-1})} = e^{(n-1)f} \left( \frac{\int _X \omega _u^n}{\int _X e^f \omega _0^n} \right) ^{n-1}, \end{aligned}$$

(2.20)

where $\omega _u$ is a positive $(1,1)$-form on $X$ such that

$$\begin{aligned} \omega _u^{n-1} = \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u\wedge \eta ^{n-2} > 0. \end{aligned}$$

Assume that $\eta$ has nonnegative orthogonal bisectional curvature. Then, we have

$$\begin{aligned} \Delta _{\eta } u \le C + C (u - \inf _X u) \quad \mathrm{on}\,X, \end{aligned}$$

(2.21)

and

$$\begin{aligned} \sup _X |\omega _u^{n-1}|_{\eta } \le C + C (\sup _X u - \inf _X u), \end{aligned}$$

where $C> 0$ is a constant depending only on $f$, $\eta$, $n$, and $\omega _0$.

Proof

Let

$$\begin{aligned} F = (n-1) \left( f + \log \int _X \omega _u^n - \log \int _X e^f \omega _0^n \right) \!. \end{aligned}$$

To apply Lemma 3, it suffices to estimate $\inf (\Delta _{\eta } F)$ and $\sup F$. Note that

$$\begin{aligned} \Delta _{\eta } F = (n-1) \Delta _{\eta } f. \end{aligned}$$

Applying the maximum principle to (2.20) at the points where $u$ attain its maximum and minimum, respectively, yields a uniform bound for the constant:

$$\begin{aligned} - \sup f \le \log \int _X \omega _u^n - \log \int _X e^f \omega _0^n \le - \inf f. \end{aligned}$$

This implies that $\sup |F| \le (n-1) (\sup f - \inf f)$. $\square$

3 $C^0$ estimates

In this section, we will derive the following general $C^0$ estimate. This together with Corollary 4 will settle the $C^0$ estimate for manifolds of nonnegative orthogonal bisectional curvature.

Lemma 5

Let $(X,\eta )$ be an arbitrary Kähler manifold with complex dimension $n \ge 2$. Suppose that $u \in C^2(X)$ satisfies

$$\begin{aligned} \Delta u&\le C_1 + C_1 (u - \inf _X u), \\ \Delta u&> - C_2, \end{aligned}$$

where $\Delta$ stands for the Laplacian with respect to $\eta$, and $C_1, C_2$ are two positive constants. Then,

$$\begin{aligned} \sup _X u - \inf _X u \le C, \end{aligned}$$

in which $C>0$ is a constant depending only on $\eta$, $n$, $C_1$, and $C_2$.

The proof is based on the following maximum principle (Proposition 6) and the weak Harnack inequality (Proposition 7). We denote for $p > 0$,

$$\begin{aligned} \Vert h\Vert _p = \left( \int h^p \eta ^n \right) ^{1/p}, \quad \mathrm{for\, all} \quad h \in L^p(X,\eta ). \end{aligned}$$

Proposition 6

Let $v \in C^2(X)$, $v > 0$ on $X$, satisfy that

$$\begin{aligned} \Delta v + c v \ge d \quad \mathrm{on}\,X, \end{aligned}$$

(3.1)

where $c$ and $d$ are constants. Then, for any real number $p > 0$,

$$\begin{aligned} \sup _X v \le C^{1/p} (1 + |c|)^{n/p} ( \Vert v\Vert _p + |d|), \end{aligned}$$

where $C>0$ is a constant depending only on $\eta$ and $n$.

Proposition 7

Let $v \in C^2(X)$, $v > 0$ on $X$ and satisfy

$$\begin{aligned} \Delta v - c v \le 0 \quad \mathrm{on} \quad X, \end{aligned}$$

(3.2)

where $c$ is a constant. Then, there exists a real number $p_0 > 0$, depending on $\eta$, $n$, and $c$, such that

$$\begin{aligned} \inf _X v \ge C^{-1/p_0} (1 + |c|)^{-n/p_0} \Vert v \Vert _{p_0}, \end{aligned}$$

where $C> 0$ depends only on $\eta$ and $n$.

Proposition 6 and Proposition 7 can be proved by Moser’s iteration. The arguments are standard (see, for example, [6]). We are in a position to prove Lemma 5.

Proof of Lemma 5

Let

$$\begin{aligned} v = u - \inf _X u + 1. \end{aligned}$$

Since $X$ is compact, $u$ attains its infimum. Then,

$$\begin{aligned} v \ge 1, \quad \mathrm{and} \quad \inf _X v = 1. \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \Delta v - C_1 v \le 0, \end{aligned}$$

(3.3)

and

$$\begin{aligned} \Delta v > - C_2. \end{aligned}$$

(3.4)

Applying Proposition 7 to (3.3), we obtain that

$$\begin{aligned} \inf _X v \ge C^{-1/p_0} ( 1 + |C_1|)^{-n/p_0} \Vert v\Vert _{p_0}. \end{aligned}$$

Here $p_0 > 0$ is a number depending only on $\eta$, $n$, and $C_1$; $C > 0$ is a constant depending only on $\eta$ and $n$. Applying Proposition 6 to (3.4) with $p = p_0$ yields that

$$\begin{aligned} \sup _X v \le (C')^{1/p_0} ( \Vert v\Vert _{p_0} + C_2), \end{aligned}$$

where $C'>0$ depends only on $\eta$ and $n$. Combining these two inequalities we have

$$\begin{aligned} \sup _X v&\le (C')^{1/p_0} \left[ C^{1/p_0} ( 1 + |C_1|)^{n/p_0} \inf _X v + C_2 \right] \\&= (C')^{1/p_0} \left[ C^{1/p_0} ( 1 + |C_1|)^{n/p_0} + C_2 \right] . \end{aligned}$$

It follows that

$$\begin{aligned} \sup _X u - \inf _X u \le \sup _X v \le C, \end{aligned}$$

where $C>0$ depends only on $\eta$, $n$, $C_1$, and $C_2$. $\square$

Let us now return to Eq. (1.4). We let $(X, \eta )$ be the complex $n$-dimensional Kähler manifold of nonnegative orthogonal bisectional curvature, and $\omega _0$ be a Hermitian metric on $X$.

Corollary 8

For any $f \in C^{\infty }(X)$, let $u \in C^{\infty }(X)$ be a solution of

$$\begin{aligned} \frac{\det (\omega _u^{n-1})}{\det (\omega _0^{n-1})} = e^{(n-1)f} \left( \frac{\int _X \omega _u^n}{\int _X e^f \omega _0^n} \right) ^{n-1} , \end{aligned}$$

where $\omega _u$ is a positive $(1,1)$-form such that

$$\begin{aligned} \omega _u^{n-1} = \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u\wedge \eta ^{n-2} > 0 \quad \mathrm{on}\,X. \end{aligned}$$

Then,

$$\begin{aligned} \sup _X |\omega _u^{n-1}|_{\eta } \le C, \end{aligned}$$

where $C> 0$ is a constant depending only on $f$, $\eta$, $n$, and $\omega _0$.

Proof

By Corollary 4, it suffices to estimate $(\sup u - \inf u)$. Contracting

$$\begin{aligned} \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u\wedge \eta ^{n-2} > 0 \end{aligned}$$

with $\eta$ yields that

$$\begin{aligned} \Delta _{\eta } u > - \frac{n \eta \wedge \omega _0^{n-1}}{\eta ^n} > - C_2 \qquad \mathrm{on}\,X. \end{aligned}$$

Here the constant $C_2 > 0$ depends only on $\eta$, $n$, and $\omega _0$. We have (2.21), on the other hand. Therefore, the result is an immediate consequence of Lemma 5. $\square$

4 Hölder estimates for second derivatives

Let $X$ be a $n$-dimensional Kähler manifold, $\eta$ be a Kähler metric on $X$, and $\omega _0$ be a balanced metric on $X$. We will establish the following estimate.

Lemma 9

For $F \in C^2(X)$, let $u \in C^4(X)$ satisfy that

$$\begin{aligned} \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u \wedge \eta ^{n-2} > 0 \qquad \mathrm{on} \,X, \end{aligned}$$

and that

$$\begin{aligned} \det [\omega _0^{n-1}+(\sqrt{-1}/2) \partial \bar{\partial }u\wedge \eta ^{n-2}]= e^{F} \det \omega _0^{n-1}. \end{aligned}$$

(4.1)

Suppose that

$$\begin{aligned} \sup _X |\omega _0^{n-1} + \sqrt{-1}/2\partial \bar{\partial } u \wedge \eta ^{n-2}|_{\eta } \le C_3 \end{aligned}$$

(4.2)

for some constant $C_3>0$. Then,

$$\begin{aligned} \Vert u \Vert _{C^{2,\alpha }(X)} \le C, \end{aligned}$$

where $0 < \alpha < 1$ and $C>0$ are constants depending only on $C_3$, $n$, $\omega _0$, and $\eta$.

We shall apply the Evans–Krylov theory (see, for example, Gilbarg–Trudinger [3, p. 461, Theorem 17.14]), which is on the real fully nonlinear elliptic equation. Note that Evans–Krylov theory is based on the weak Harnack estimate (see, for example, [3, p. 246, Theorem 9.22]), which, in turn, makes uses of Aleksandrov’s maximum principle (see, for example, [3, p. 222, Lemma 9.3]).

We first adapt Aleksandrov’s maximum principle to the complex setting. To see this, we start from the following result (see, for example, Lemma 9.2 in [3]): Let $\Omega \subset {\mathbb {C}}^n$ be a bounded domain with smooth boundary.

Lemma

(Aleksandrov) For $v \in C^2(\overline{\Omega })$ with $v \le 0$ on $\partial \Omega$, we have

$$\begin{aligned} \sup _{\Omega } v \le \frac{\mathrm{diam}(\Omega )}{\sigma _{2n}^{1/(2n)}} \left( \,\, \int _{\Gamma ^+_v} |\det D^2 v| \right) ^{\frac{1}{2n}}. \end{aligned}$$

(4.3)

Here $\sigma _{2n}$ is the volume of unit ball in ${\mathbb {C}}^n$, $D^2 v$ denotes the real Hessian matrix of $v$, and $\Gamma ^+_v$ is the upper contact set of $v$, i.e.,

$$\begin{aligned} \Gamma _v^+ = \{ y \in \Omega ; v(x) \le v(y) + Dv(y)\cdot (x - y) \quad \mathrm{for\,all} \quad x \in \Omega \}. \end{aligned}$$

Then, it suffices to control the real Hessian $D^2 v$ by the complex Hessian $(v_{i\bar{j}})$ of $v$, over $\Gamma _v^+$. Note that $\Gamma _v^+ \subset \{ y \in \Omega ; (D^2 v) (y) \le 0 \}$. We shall make use of the following inequality (comparing with [1, p. 246], we do not need Hadarmad’s inequality for semipositive matrices):

Proposition 10

Let $w$ be a real $C^2$ function in $\Omega$. For $P \in \Omega$ such that $D^2 w \ge 0$,

$$\begin{aligned} \det (D^2 w) \le 8^n |\det w_{i\bar{j}} |^2 \quad \mathrm{at}\,P. \end{aligned}$$

Proof

Recall that

$$\begin{aligned} \frac{\partial }{\partial z^i} = \frac{1}{2}\left( \frac{\partial }{\partial x^i} - \sqrt{-1} \frac{\partial }{\partial y^i} \right) , \quad 1 \le i \le n. \end{aligned}$$

We denote

$$\begin{aligned} w_{x^i} = \frac{\partial w}{\partial x^i}, \quad w_{x^i y^j} = \frac{\partial ^2 w}{\partial x^i \partial y^j}, \quad \ldots . \end{aligned}$$

Then,

$$\begin{aligned} w_{i\bar{j}} = \frac{1}{4} \left( w_{x^i x^j} + w_{y^i y^j} \right) + \frac{\sqrt{-1}}{4} \left( w_{x^i y^j} - w_{x^j y^i} \right) , \quad 1 \le i, j \le n. \end{aligned}$$

Since $D^2 w \ge 0$ at $P$, we can choose a coordinate system $(x^1,y^1, \ldots , x^n, y^n)$ near $P$ such that $D^2 w$ is diagonalized at $P$, and hence,

$$\begin{aligned} w_{x^i x^i} \ge 0, \quad w_{y^i y^i} \ge 0, \quad \mathrm{for\,all} \quad 1 \le i \le n. \end{aligned}$$

Then, under this coordinate system, the complex Hessian of $w$ is also diagonalized, i.e.,

$$\begin{aligned} w_{i\bar{j}} = \frac{\delta _{ij}}{4} \left( w_{x^i x^i} + w_{y^i y^i} \right) . \end{aligned}$$

It follows that, at $P$,

$$\begin{aligned} 16^n |\det w_{i\bar{j}} |^2&= \prod _{i=1}^n \left( w_{x^i x^i} + w_{y^i y^i} \right) ^2 \\&\ge 2^n \prod _{i=1}^n w_{x^i x^i} \prod _{i=1}^n w_{y^i y^i} \\&= 2^n \det (D^2 w). \end{aligned}$$

Moreover, for any Hermitian matrix $(a^{i\bar{j}}) > 0$ on $\Gamma _v^+$, we have by an elementary inequality that

$$\begin{aligned} \det (a^{i\bar{j}}) \det (- v_{i\bar{j}}) \le \left( \frac{- \sum _{i,j} a^{i\bar{j}} v_{i\bar{j}} }{n} \right) ^n. \end{aligned}$$

(4.4)

Now apply Proposition 10 and (4.4) to (4.3) to obtain the following complex version Aleksandrov’s maximum principle (compare with [3, p. 222, Lemma 9.3]):

Lemma 11

Let $(a^{i\bar{j}})$ be a positive definite Hermitian matrix in $\Omega$. For $v \in C^2(\overline{\Omega })$ with $v \le 0$ on $\partial \Omega$,

$$\begin{aligned} \sup _{\Omega } v \le \frac{2^n \mathrm{diam}(\Omega )}{ n\,\sigma _{2n}^{1/(2n)}} \left[ \,\int _{\Gamma _v^+} \Big |\frac{- \sum a^{i\bar{j}} v_{i\bar{j}} }{\det (a_{i\bar{j}})^{1/n}} \Big |^{2n} \right] ^{\frac{1}{2n}}. \end{aligned}$$

Then, the weak Harnack inequality below (compare with [3, p. 246, Theorem 9.22]) follows from Lemma 11 and the cube decomposition procedure.

Theorem

(Krylov–Safonov) Let $v \in W^{2,2n}(\Omega )$ satisfy $\sum a^{i\bar{j}} v_{i\bar{j}} \le g$ in $\Omega$, where $g \in L^{2n}(\Omega )$, and $(a^{i\bar{j}})$ satisfies that

$$\begin{aligned} 0 < \lambda |\zeta |^2 \le \sum _{i,j} a^{i\bar{j}}(z) \zeta _i \zeta _j \le \Lambda |\zeta |^2, \quad \mathrm{for\,all} \quad z \in \Omega \,\mathrm{and}\,\zeta \in {\mathbb {C}}^n, \end{aligned}$$

in which $\lambda$ and $\Lambda$ are two constants. Suppose that $v \ge 0$ in an open ball $B_{2R}(y) \subset \Omega$ centered at $y$ of radius $2R$. Then,

$$\begin{aligned} \left( \frac{1}{|B_R|} \int _{B_R} v^p \right) ^{1/p} \le C \left[ \inf _{B_R} v + \frac{R}{\lambda } \Vert g \Vert _{L^{2n}(B_{2R})} \right] , \end{aligned}$$

where $|B_R|$ denotes the measure of $B_R$, and $p > 0$ and $C > 0$ are constants depending only on $n$, $\lambda$, and $\Lambda$.

Let us denote by

$$\begin{aligned} E[(u_{i\bar{j}})] = \log \det \left[ \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u \wedge \eta ^{n-2} \right] . \end{aligned}$$

To apply Evans–Krylov theory, it remains to check the following two conditions ([3, p. 456]):

• (1) $E$ is uniformly elliptic with respect to $(u_{i\bar{j}})$,

• (2) $E$ is concave on the range of $(u_{i\bar{j}})$.

As in Sect. 2, we denote $\Psi = \omega _0^{n-1}$ and

$$\begin{aligned} \Psi _u = \Psi + (\sqrt{-1}/2) \partial \bar{\partial }u \wedge \eta ^{n-2}. \end{aligned}$$

(4.5)

We use the index convention (2) for an $(n-1, n-1)$-form. Then,

$$\begin{aligned} E[ (u_{i\bar{j}}) ] = \log \det [(\Psi _u)_{i\bar{j}}], \end{aligned}$$

and thus,

$$\begin{aligned} \frac{\partial E}{\partial (\Psi _u)_{i\bar{j}}} = (\Psi _u)^{i\bar{j}}, \quad \frac{\partial ^2 E}{\partial (\Psi _u)_{i\bar{j}} \partial (\Psi _u)_{k\bar{l}}} = - (\Psi _u)^{i\bar{l}} (\Psi _u)^{k\bar{j}}. \end{aligned}$$

Clearly, $E$ is concave on $[(\Psi _u)_{i\bar{j}}]$. By (4.1) and (4.2), we know that the eigenvalues of $[(\Psi _u)_{i\bar{j}}]$ with respect to $(\eta _{i\bar{j}})$, have uniform bounds which depend only on $F$, $\omega _0$, and $C_3$. Therefore, $E$ is uniformly elliptic with respect to $[(\Psi _u)_{i\bar{j}}]$. Observe that by (4.5), $[(\Psi _u)_{i\bar{j}}]$ depends linearly on $(u_{p\bar{q}})$. Since $(\eta _{k\bar{l}}) > 0$ on $X$, the conditions (1) and (2) follows immediately from the chain rule.

Now we can apply the procedure in [3, p. 457–461], and this proves Lemma 9. As a corollary, we obtain the Hölder estimate of $C^2$ for Eq. (1.4).

Corollary 12

Let $(X,\eta )$ an $n$-dimensional Kähler of nonnegative quadratic bisectional curvature, and $\omega _0$ be a Hermitian metric on $X$. Given any $f \in C^{\infty }(X)$, let $u \in C^{\infty }(X)$ be a solution of

$$\begin{aligned} \frac{\det (\omega _u^{n-1})}{\det (\omega _0^{n-1})} = e^{(n-1)f} \left( \frac{\int _X \omega _u^n}{\int _X e^f \omega _0^n} \right) ^{n-1} , \end{aligned}$$

where $\omega _u$ is a positive $(1,1)$-form such that

$$\begin{aligned} \omega _u^{n-1} = \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u\wedge \eta ^{n-2} > 0 \quad \mathrm{on} \quad X. \end{aligned}$$

Then,

$$\begin{aligned} \Vert u \Vert _{C^{2,\alpha }(X)} \le C, \end{aligned}$$

where $0 < \alpha < 1$ and $C> 0$ are constants depending only on $f$, $\eta$, $n$, and $\omega _0$.

5 Openness and uniqueness

Throughout this section, we let $\omega _0$ be a balanced metric, and let $\eta$ be an arbitrary Kähler metric, unless otherwise indicated. We fix $k \ge n+4$, $0 < \alpha < 1$, and a function $f \in C^{k,\alpha }(X)$ satisfying

$$\begin{aligned} \int _X e^f \omega _0^n = V \equiv \int _X \omega _0^n. \end{aligned}$$

Here $C^{k,\alpha }(X)$ is the usual Hölder space on $X$. Consider for $0 \le t \le 1$,

$$\begin{aligned} \frac{\det (\omega _{u_t}^{n-1})}{\det (\omega _0^{n-1}) } = e^{(n-1)tf} \left( \frac{\int _X \omega _{u_t}^n}{\int _X e^{tf} \omega _0^n} \right) ^{n-1}, \end{aligned}$$

(5.1)

where $u_t \in {\mathcal {P}}_{\eta }(\omega _0)$. By abuse of notation, in this section we denote

$$\begin{aligned} {\mathcal {P}}_{\eta } (\omega _0) = \left\{ v \in C^{k+2,\alpha }(X); \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }v \wedge \eta ^{n-2} > 0\right\} . \end{aligned}$$

Let

$$\begin{aligned} T = \{ t\in [0,1] ;&\mathrm{the\,equation (5.1)\,has\,a\,solution} \quad u_t \in C^{k+2,\alpha }(X) \nonumber \\&\mathrm{such\,that}\, u_t \in {\mathcal {P}}_{\eta }(\omega _0). \}. \end{aligned}$$

(5.2)

Clearly, we have $0 \in T$.

Lemma 13

Let $T$ be the set given as above. Then $T$ is open in $[0,1]$.

Proof

Notice that (5.1) is the same as

$$\begin{aligned} \frac{\omega _{u_t}^n}{\omega _0^n} = e^{tf} \frac{\int _X \omega _{u_t}^n}{\int _X e^{tf} \omega _0^n}. \end{aligned}$$

As in Section 3 of [2], we define

$$\begin{aligned} M (w) \equiv \log \frac{\omega _{w}^n}{\omega ^n_0} - \log \left( \frac{1}{V} \int _X \omega _{w}^n \right) , \end{aligned}$$

for any $w \in {\mathcal {P}}_{\eta }(\omega _0)$. Then, $M(w) \in {\mathcal {F}}^{k,\alpha }(X)$, where ${\mathcal {F}}^{k,\alpha }(X)$ is the hypersurface in $C^{k,\alpha }(X)$ given by

$$\begin{aligned} {\mathcal {F}}^{k,\alpha }(X) = \left\{ g \in C^{k,\alpha }(X); \int _X e^g \, \omega _0^n = V \right\} . \end{aligned}$$

Now suppose that $t \in T$. Then, the corresponding $u_t$ defines a positive $(1,1)$-form $\omega _{u_t}$ such that

$$\begin{aligned} \omega _{u_t}^{n-1} = \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }u \wedge \eta ^{n-2} > 0 \quad \mathrm{on}\,X; \end{aligned}$$

furthermore, $u_t$ satisfies that

$$\begin{aligned} M(u_t) = tf + \log V - \log \left( \, \int _X e^{tf} \omega _0^n \right) \in {\mathcal {F}}^{k,\alpha }(X). \end{aligned}$$

The tangent space of ${\mathcal {F}}^{k,\alpha }(X)$ at $M(u_t)$ is identically the same as the Banach space ${\mathcal {E}}_t^{k,\alpha }(X)$, which consists of all $h \in C^{k,\alpha }(X)$ such that

$$\begin{aligned} \int _X h \, \omega _{u_t}^n = 0. \end{aligned}$$

In view of the Implicit Function Theorem, it suffices to show that the linearization operator $L_t \equiv M_{u_t}$, given by

$$\begin{aligned} L_t (v) = \frac{n(\sqrt{-1}/2)\partial \bar{\partial } v \wedge \eta ^{n-2} \wedge \omega _{u_t}}{(n-1)\omega _{u_t}^n} - \frac{n \int _X (\sqrt{-1}/2) \partial \bar{\partial }v \wedge \eta ^{n-2} \wedge \omega _{u_t}}{(n-1)\int _X \omega _{u_t}^n}, \end{aligned}$$

is a linear isomorphism from ${\mathcal {E}}_t^{k+2,\alpha }(X)$ to ${\mathcal {E}}_t^{k,\alpha }(X)$. This is guaranteed by Lemma 13 in [2]. The proof is thus finished. $\square$

Remark 14

We thank John Loftin for pointing out that the openness argument in [2] also works for $\eta$ being a astheno-Kähler metric, i.e., $\eta$ is a hermitian metric such that $\partial \bar{\partial } \eta ^{n-2} = 0$.

By the results in the previous section, we know that $T$ is also closed, provided that the orthogonal bisectional curvature of $\eta$ is nonnegative. Therefore, the existence part in Theorem 1 is proved. The uniqueness follows immediately from the following proposition.

Proposition 15

Let $v \in {\mathcal {P}}_{\eta }(\omega _0)$ satisfying

$$\begin{aligned} \det \left[ \omega _0^{n-1} + (\sqrt{-1}/2) \partial \bar{\partial }v \wedge \eta ^{n-2} \right] = \delta \det \omega _0^{n-1}, \end{aligned}$$

(5.3)

where $\delta >0$ is a constant. Then, $v$ must be a constant function and $\delta = 1$.

Proof

Applying the maximum principle to Eq. (5.3) at the maximum points of $v$ yields that $\delta \le 1$. Similarly, we get $\delta \ge 1$ by considering (5.3) at the minimum points of $v$. Thus, $\delta = 1$. Then, we apply the arithmetic–geometric mean inequality to obtain

$$\begin{aligned} 1&= \left[ \frac{\det \omega _v^{n-1}}{\det \omega _0^{n-1}}\right] ^{1/n} \le 1 + \frac{1}{n} \sum _{i,j=1}^n (\omega _0^{n-1})^{i\bar{j}} \left( (\sqrt{-1}/2) \partial \bar{\partial }v \wedge \eta ^{n-2}\right) _{i\bar{j}}\\&= 1 + \frac{\omega _0 \wedge \eta ^{n-2} \wedge (\sqrt{-1}/2) \partial \bar{\partial }v }{\omega _0^n} \equiv 1 + K v. \end{aligned}$$

Note that the linear operator $K$ so defined is uniformly elliptic, by the metric equivalence of $\eta$ and $\omega _0$ on the compact manifold $X$. Applying the strong maximum principle to $K v \ge 0$ yields that $v$ is a constant function. $\square$

Therefore, the proof of Theorem 1 is completed.