AUXILIARY MONGE-AMPERE EQUATIONS IN GEOMETRIC ANALYSIS 111Contribution to
Surveys in Differential Geometry in honor of S.S. Chern. Work supported in
part by the National Science Foundation under grant DMS-22-03273.

Bin Guo and Duong H. Phong

###### Abstract

This is an introduction to a particular class of auxiliary complex Monge-
Ampère equations which had been instrumental in $L^{\infty}$ estimates for
fully non-linear equations and various questions in complex geometry. The
essential comparison inequalities are reviewed and shown to apply in many
contexts. Adapted to symplectic geometry, with the auxiliary equation given
now by a real Monge-Ampère equation, the method gives an improvement of an
earlier theorem of Tosatti-Weinkove-Yau, reducing Donaldson’s conjecture on
the Calabi-Yau equation with a taming symplectic form from an exponential
bound to an $L^{1}$ bound.

##  1 Introduction

It is well-known that comparisons with an auxiliary equation can be of
powerful assistance in the study of partial differential equations.
Comparisons with harmonic functions were used early on by De Giorgi [17], and
have been applied since in many contexts, including e.g. in the book of L.
Simon [58], and in X.J. Wang’s method for Schauder estimates for the Poisson
equation [71]. Since then, comparisons with many other equations have proved
to be effective. Complex Monge-Ampère equations are particularly suitable as
auxiliary equations, since the existence and smoothness of their solutions for
given right hand sides have been established by Yau [72] in the case of
compact Kähler manifolds, and by Caffarelli, Kohn, Nirenberg, and Spruck [7]
in the case of the Dirichlet problem. Notable successes of Monge-Ampère
equations as auxiliary equation include the estimates of Dinew and Kolodziej
[20] relating volume and capacity, the bounds of Song and Tian [60] for the
Kähler-Ricci flow, and the entropy estimates of Chen and Cheng [9] for the
constant scalar curvature equation.

Very recently, a specific class of auxiliary Monge-Ampère equations has been
instrumental in many significant advances in complex geometry. This class was
first introduced by the authors in joint work with F. Tong in [31], and
already led to a pure PDE proof of the $L^{\infty}$ estimates of Kolodziej
[45], a goal which had eluded researchers in the field for close to a quarter
of a century. As a PDE proof, the method extends immediately Kolodziej’s
estimates to a general class of non-linear equations satisfying a structural
condition. Remarkably, this class has been shown by Harvey and Lawson [43] to
be quite large, and include in particular all invariant Garding-Dirichlet
operators. But the method turns out to be even more flexible and powerful than
naively anticipated, and it has had since many unexpected applications. These
include stability estimates for Monge-Ampère and Hessian equations [33];
$L^{\infty}$ estimates for Monge-Ampère equations on nef classes rather than
just Kähler classes [34]; sharp modulus of continuity for non-Hölder solutions
[35]; extensions to parabolic equations [10]; extensions to equations on
Hermitian manifolds [37]; extensions to form-type equations [37]; lower bounds
for the Green’s function [36]; uniform entropy estimates [38]; and diameter
estimates and convergence theorems in Kähler geometry not requiring bounds on
the Ricci curvature [40, 39].

The main purpose of this paper is to provide a survey of these developments.
We shall describe in some detail the essential features of the particular
class of auxiliary complex Monge-Ampère equations of interest. The main
applications are then sketched, with explanations of how the auxiliary Monge-
Ampère equations are used and precise statements of the results obtained. In
all but one case, the full treatment is left to references to the original
papers in the literature. The one exceptional case is the application to
taming symplectic forms on almost-Kähler manifolds. It is a conjecture of
Donaldson [22], motivated by symplectic geometry, that on a compact
$4$-manifold equipped with an almost-complex structure $J$ and a taming
symplectic form $\Omega$, the Calabi-Yau equation would admit a priori bounds
to all orders. This conjecture had been reduced by Weinkove [71] to an
$L^{\infty}$ bound for the potential, and subsequently by Tosatti, Weinkove,
and Yau [68] to a single exponential estimate. Using an auxiliary real Monge-
Ampère equation, we can reduce it further to a single $L^{1}$ estimate. This
result is treated in detail because it is new and does not appear anywhere
else. But it may also be noteworthy as evidence that the methods here can
extend to the real or symplectic context as well.

##  2 The auxiliary Monge-Ampère equation

We begin by describing the class of Monge-Ampère equations which will serve
later as comparison equations. Some key inequalities needed for the maximum
principle are broadly described, which can be adapted later for different
applications.

Let $(X,\omega_{X})$ be a compact $n$-dimensional Hermitian manifold. For any
Hermitian form $\omega$ on $X$ and any smooth function $\varphi$ with ${\rm
sup}_{X}\varphi=0$, set
$\omega_{\varphi}=\omega+i\partial\bar{\partial}\varphi$, and consider the
relative endomorphism

$\displaystyle h_{\varphi}=\omega_{X}^{-1}\omega_{\varphi}.$ (2.1)

We denote by $\lambda[h_{\varphi}]$ the (un)-ordered vector of its
eigenvalues. Let $f(\lambda)$ be a function on a convex cone
$\Gamma\subset{\bf R}^{n}$ invariant under permutations, and consider the
family of equations parametrized by $\omega$,

$\displaystyle
f(\lambda[h_{\varphi}])=k_{\omega}(z),\quad\lambda[h_{\varphi}]\in\Gamma,$
(2.2)

where $k_{\omega}(z)$ is a given positive function. For some estimates, it is
convenient to introduce the constant $c_{\omega}>0$ defined by the following
normalization

$\displaystyle
k(z)=c_{\omega}\,e^{F_{\omega}},\quad\int_{X}e^{nF_{\omega}}\omega_{X}^{n}=\int_{X}\omega_{X}^{n}.$
(2.3)

As in [31], we require that $f:\Gamma\to{\bf R}_{+}$ satisfies

(1) $\Gamma\subset{\bf R}^{n}$ is a symmetric cone with

$\Gamma_{n}\subset\Gamma\subset\Gamma_{1}.$ (2.4)

Here $\Gamma_{k}$ is the cone of vectors $\lambda$ with
$\sigma_{j}(\lambda)>0$ for $1\leq j\leq k$, where $\sigma_{j}(\lambda)$ is
the $j$-th symmetric polynomial in $\lambda$. In particular, $\Gamma_{1}$ is
the half-space defined by $\lambda_{1}+\cdots+\lambda_{n}>0$, and $\Gamma_{n}$
is the first octant, defined by $\lambda_{j}>0$ for $1\leq j\leq n$.

(2) $f(\lambda)$ is symmetric in
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\Gamma$ and it is homogeneous of
degree one;

(3) $\frac{\partial f}{\partial\lambda_{j}}>0$ for each $j=1,\ldots,n$ and
$\lambda\in\Gamma$;

(4) There is a $\gamma>0$ such that

$\prod_{j=1}^{n}\frac{\partial f}{\partial\lambda_{j}}\geq\gamma\,{\rm
det}^{-1}(\omega_{X})\,\quad\forall\lambda\in\Gamma.$ (2.5)

It is well-known that equations such as the Monge-Ampère equation, with
$f(\lambda)=(\prod_{j=1}^{n}\lambda_{j})^{1\over n}$, or the Hessian equation
with $f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$ where $\sigma_{k}$ is the
$k$-th order symmetric polynomial, or the $p$-Monge-Ampère equation of Harvey
and Lawson [41, 42] with

$f(\lambda)=\Big{(}\prod_{I}\lambda_{I}\Big{)}^{\frac{n!}{(n-p)!p!}}$

where $I$ runs over all distinct multi-indices
$1\leq{i_{1}}<\cdots<{i_{p}}\leq n$,
$\lambda_{I}=\lambda_{i_{1}}+\cdots+\lambda_{i_{p}}$, and $\Gamma$ is the cone
defined by $\lambda_{I}>0$ for all $p$-indices $I$, all satisfy the structural
condition (4). A remarkable recent result of Harvey and Lawson [43] is that
the condition (4) actually holds for very large classes of non-linear
operators, including all invariant Garding-Dirichlet operators. As noted in
[43], the condition (4) also arose independently in [2] in the study of
$W^{2,p}$ interior regularity.

We would like to compare the solution $\varphi$ of the equation (2.2) with the
solution $\psi$ of the following complex Monge-Ampère equation

$\displaystyle(\omega+i\partial\bar{\partial}\psi)^{n}={\tau(-\varphi+q(z)-s)\over
A}k_{\omega}^{n}(z)\,\omega_{X}^{n}.\qquad{\rm sup}_{X}\psi=0$ (2.6)

where $s\geq 0$ is a nonnegative constant, $\tau(t)$ is a smooth strictly
positive function, $q(z)$ is a given function, and $A$ is a normalizing
constant.

Let $G^{j\bar{k}}$ be the linearized operator of $\,{\rm log}\,f(\lambda)$,
defined by

$\displaystyle G^{j\bar{k}}={\partial\over\partial h_{\bar{k}j}}\,{\rm
log}\,f(\lambda[h])$ (2.7)

Fix a point $z_{0}\in X$. Since $\omega_{X}$ is positive definite, and
$h_{\varphi}$ is a self-adjoint endomorphism with respect to $\omega_{X}$, we
can choose a holomorphic coordinate system centered at $z_{0}$ where
$(\omega_{X})_{\bar{j}k}=\delta_{jk}$ and $h_{\varphi}(z_{0})$ is diagonal,
$(h_{\varphi})_{\bar{j}k}(z_{0})=\lambda_{j}\delta_{jk}$. In particular
$(\omega_{\varphi})_{\bar{k}j}(z_{0})=\lambda_{j}\delta_{jk}$.

###### Lemma 1

The linearized endomorphism $G^{j\bar{k}}$ then satisfies the following: for
$\lambda\in\Gamma$

(a) $G^{j\bar{k}}(z_{0})={1\over f(\lambda)}{\partial
f\over\partial\lambda_{j}}\delta_{jk}$, and $G^{j\bar{k}}$ is positive
definite;

(b) $G^{j\bar{k}}(\omega_{\varphi})_{\bar{k}j}(z_{0})=1$ and
$G^{j\bar{k}}\omega_{\bar{k}j}(z_{0})\geq 0$,

(c) and, taking into account the equations satisfied by $\varphi$ and $\psi$,

$\displaystyle{1\over
n}G^{j\bar{k}}(\omega_{\psi})_{\bar{k}j}\geq({\gamma\tau(-\varphi+q-s)\over
A})^{1\over n}.$ (2.8)

Proof. The formula in (a) is a classic formula for the linearization of a
fully non-linear operator $\,{\rm log}\,f(\lambda[h])$ at a diagonal matrix
$h$ from the theory of non-linear equations [61]. The positive-definiteness of
$G^{j\bar{k}}$ is an immediate consequence of the ellipticity condition
$\partial_{j}f(\lambda)>0$ of the function $f$. Next, we can write at $z_{0}$,

$\displaystyle G^{j\bar{j}}(\omega_{\varphi})_{\bar{k}j}=\sum_{j=1}^{n}{1\over
f(\lambda)}{\partial f\over\partial\lambda_{j}}\lambda_{j}=1$ (2.9)

by Euler’s relation for homogeneous functions $f(\lambda)$ of degree $1$. This
proves the first equation in (b). To establish the second equation in (b), we
observe that since $(\omega_{X})_{\bar{k}j}=\delta_{jk}$ at $z_{0}$, $\omega$
can be identified with the relative endomorphism
$h_{0}=(\omega_{X})^{-1}\omega$, which is hermitian and positive with respect
to $\omega_{X}$. The expression $G^{j\bar{k}}\omega_{\bar{k}j}$ can then be
identified with the trace ${\rm Tr}(Gh_{0})$, which is then the inner product
of two positive matrices. As such, it is positive, as can be seen for example
by writing it in a basis where both endomorphisms are diagonal.

Finally, to establish (c), we apply the arithmetic-geometric inequality to
write

$\displaystyle{1\over n}G^{j\bar{k}}(\omega_{\psi})_{\bar{k}j}$
$\displaystyle\geq$ $\displaystyle\bigg{(}{\rm
det}\,(G^{j\bar{k}}(\omega_{\psi})_{\bar{k}m})\bigg{)}^{1\over n}=\bigg{(}{\rm
det}G^{j\bar{k}}\cdot{\rm det}(\omega_{\psi})_{\bar{k}m})\bigg{)}^{1\over n}$
(2.10) $\displaystyle=$ $\displaystyle\bigg{(}{1\over
f(\lambda)^{n}}(\prod_{j=1}^{n}{\partial
f\over\partial\lambda_{j}})\cdot{\tau(-\varphi+q-s)\over
A}k_{\omega}(z)^{n}{\rm det}(\omega_{X})\bigg{)}^{1\over n}.$

Applying the equation for $\varphi$, this inequality simplifies to

$\displaystyle{1\over
n}G^{j\bar{k}}(\omega_{\psi})_{\bar{k}j}\geq\big{(}\prod_{j=1}^{n}{\partial
f\over\partial\lambda_{j}}\cdot{\tau(-\varphi+q-s)\over A}{\rm
det}(\omega_{X})\big{)}^{1\over n}$ (2.11)

Applying now the structural condition on $f(\lambda)$, we obtain the desired
inequality (c). Q.E.D.

Next, we need a “comparison function $\Phi$”, relating the solution $\varphi$
of the equation (2.2) to the solution $\psi$ of the auxiliary Monge-Ampère
equation (2.6). For the applications considered in this survey, the comparison
function $\Phi$ is usually of the form

$\displaystyle\Phi=-\varepsilon(-\psi+q(z)+\Lambda)^{b}-\varphi+\tilde{q}(z)-s$
(2.12)

with $0<b<1$ a fixed constant, $\varepsilon,\Lambda$ non-negative constants to
be chosen later, and $\tilde{q}(z)$ a smooth function. We obtain the key
inequality relating $\varphi$ and $\psi$ if we can choose all the data in
$\Phi$ so as to guarantee that $\Phi\leq 0$ everywhere. To do so, we typically
apply the maximum principle, and make use of the following general
calculations:

###### Lemma 2

Fix a point $z_{0}$ and a holomorphic coordinate system centered at $z_{0}$ as
above. Then we have the following inequality at $z_{0}$

$\displaystyle G^{j\bar{k}}\Phi_{\bar{k}j}$ $\displaystyle\geq$
$\displaystyle\varepsilon
nb(-\psi+q(z)+\Lambda)^{b-1}({\gamma\tau(-\varphi+q-s)\over A})^{1\over n}-1$
(2.13) $\displaystyle+G^{j\bar{k}}\big{\\{}\omega_{\bar{k}j}+(1-\varepsilon
b(-\psi+q+\Lambda)^{b-1})(\omega_{q})_{\bar{k}j}+\tilde{q}_{\bar{k}j}\big{\\}}.$

Proof. A direct calculation gives

$\displaystyle\Phi_{\bar{k}j}$ $\displaystyle=$ $\displaystyle\varepsilon
b(-\psi+q(z)+\Lambda)^{b-1}(\psi-q)_{\bar{k}j}-\varphi_{\bar{k}j}+\tilde{q}_{\bar{k}j}$
(2.14) $\displaystyle-\varepsilon
b(b-1)(-\psi+q(z)+\Lambda)^{b-2}(\psi-q)_{j}(\psi-q)_{\bar{k}}$

Rewriting this expression using
$(\psi-q)_{\bar{k}j}=(\omega_{\psi})_{\bar{k}j}-(\omega_{q})_{\bar{k}j}$ and
$\varphi_{\bar{k}j}=(\omega_{\varphi})_{\bar{k}j}-\omega_{\bar{k}j}$, we
obtain

$\displaystyle G^{j\bar{k}}\Phi_{\bar{k}j}$ $\displaystyle=$
$\displaystyle\varepsilon
b(-\psi+q+\Lambda)^{b-1}G^{j\bar{k}}(\omega_{\psi})_{\bar{k}j}-\varepsilon
b(-\psi+q+\Lambda)^{b-1}G^{j\bar{k}}(\omega_{q})_{\bar{k}j}$ (2.15)
$\displaystyle-G^{j\bar{k}}(\omega_{\varphi})_{\bar{k}j}+G^{j\bar{k}}\omega_{\bar{k}j}+G^{j\bar{k}}\tilde{q}_{\bar{k}j}$
$\displaystyle-\varepsilon
b(b-1)(-\psi+q+\Lambda)^{b-2}G^{j\bar{k}}(\psi-q)_{j}(\psi-q)_{\bar{k}}$

Since $G^{j\bar{k}}>0$ and $0<b<1$, we can drop from the right hand side the
expression involving $G^{j\bar{k}}(\psi-q)_{j}(\psi-q)_{\bar{k}}$. The desired
inequality follows then from the inequalities in Lemma 1. Q.E.D.

The lemma is particularly useful when the last term on the right hand side of
(2.13) happens to be positive and can be dropped. We obtain then an upper
bound for the solution $\varphi$ of the given equation (2.2) in terms of the
solution $\psi$ of the auxiliary Monge-Ampère equation. The following is the
simplest illustration, which will be shown later to apply to $L^{\infty}$
estimates on Kähler manifolds:

###### Lemma 3

Let $\varphi$ and $\psi$ satisfy the equations (2.2) and (2.6), under the
preceding hypotheses on the operator $f(\lambda)$. Assume that the function
$\tau(t)$ satisfies the condition

$\displaystyle\tau(t)\geq t^{a},\quad t\in[0,\infty)$ (2.16)

for some fixed power $a>0$. Then for any $s\geq 0$, we have

$\displaystyle-\varphi-s\leq c_{n,a,\gamma}A^{1\over
a+n}(-\psi+\Lambda)^{n\over n+a}$ (2.17)

for all $z\in X$ and all $s\geq 0$, if the constants $b$, $\varepsilon$ and
$\Lambda$ are chosen to be

$\displaystyle b={n\over n+a},\quad\varepsilon=(nb\gamma^{1\over n})^{-{n\over
a+n}}A^{1\over a+n},\quad\Lambda^{1-b}=\varepsilon b.$ (2.18)

Here $c_{n,a,\gamma}$ is a constant depending only on $n,a,\gamma$.

Proof. Let $\Phi$ be defined as in (2.12), with $q(z)=\tilde{q}(z)=0$. We
apply Lemma 2. Since $q=\tilde{q}=0$, we have $\omega_{q}=\omega$ and the last
expression on the right hand side of (2.13) reduces to

$\displaystyle G^{j\bar{k}}\big{\\{}\omega_{\bar{k}j}+(1-\varepsilon
b(-\psi+q+\Lambda^{b-1}))(\omega_{q})_{\bar{k}j}+\tilde{q}_{\bar{k}j}\big{\\}}$
$\displaystyle=$ $\displaystyle G^{j\bar{k}}\big{\\{}1-\varepsilon
b(-\psi+\Lambda)^{b-1}\big{\\}}\omega_{\bar{k}j}$ (2.19) $\displaystyle\geq$
$\displaystyle(1-\varepsilon b\Lambda^{b-1})G^{j\bar{k}}\omega_{\bar{k}j}\geq
0$

since $-\psi\geq 0$, $0<b<1$, and $G^{j\bar{k}}\omega_{\bar{k}j}\geq 0$ by (b)
of Lemma 1.

Let $z_{0}$ be a point where $\Phi$ attains its maximum on $X$. We shall show
that $\Phi(z_{0})\leq 0$. If $-\varphi(z_{0})-s\leq 0$, the function $\Phi$ is
manifestly $\leq 0$ at its maximum $z_{0}$, and we are done. Otherwise, we
note that $0\geq G^{j\bar{k}}\Phi_{\bar{k}j}(z_{0})$ and apply Lemma 2. As
just noted, we can drop the last term on the right hand side of (2.13), and
bound $\tau(-\varphi-s)$ from below by $(-\varphi-s)^{a}$. We find

$\displaystyle 1\geq n\varepsilon
b(-\psi+\Lambda)^{b-1}({\gamma(-\varphi-s)^{a}\over A})^{1\over n}$ (2.20)

at $z_{0}$, which can be rewritten as

$\displaystyle-(nb\gamma^{1\over n})^{-{n\over a+n}}A^{1\over
a+n}(-\psi+\Lambda)^{n\over n+a}-\varphi-s\leq 0.$ (2.21)

With the choice of $b$, $\varepsilon$, and $\Lambda$ indicated in the lemma,
we can recognize the left hand side as $\Phi(z_{0})$. Since $z_{0}$ is a
maximum for $\Phi$, it follows that $\Phi(z)\leq 0$ for any $z\in X$. This
last statement can be recast in the form stated in the Lemma. Q.E.D.

##  3 Application to the compact Kähler case

The first application that we discuss is the one where the above class of
auxiliary equations was originally introduced [31], in order to provide a PDE
proof of the $L^{\infty}$ estimates for the complex Monge-Ampère equation
originally established by Kolodziej [45]. We discuss this case in some detail,
since it also serves as a template for other subsequent applications.

Assume in this section that $X$ is a compact Kähler manifold, and both
$\omega_{X}$ and $\omega$ are Kähler forms. We would like to derive
$L^{\infty}$ estimates for the solutions $\varphi$ of the equation (2.2). Our
goal is to find in the present case of non-linear equations an adaptation of
the strategy going back to De Giorgi for $L^{\infty}$ bounds for linear
equations in divergence form. In this strategy, a lower bound for $\varphi$ is
obtained by showing that the set

$\displaystyle\Omega_{s}=\\{\varphi<-s\\}$ (3.1)

is empty starting from some $S_{0}$ which can be estimated. We shall deduce
this from suitable growth conditions on the function

$\displaystyle\phi(s)={1\over
V_{\omega}}\int_{\Omega_{s}}k_{\omega}^{n}(z)\omega_{X}^{n},\quad
V_{\omega}=\int_{X}\omega^{n},$ (3.2)

which will follow themselves from a reverse Hölder inequality for the key
function $A_{s}$ defined for $s>0$ by

$\displaystyle A_{s}={1\over
V_{\omega}}\int_{\Omega_{s}}(-\varphi-s)k_{\omega}^{n}(z)\omega_{X}^{n}.$
(3.3)

For this we need an auxiliary Monge-Ampère equation. Let $\tau_{\ell}(t)$ a
sequence of smooth strictly positive functions on ${\bf R}$ which decreases
monotonically to the function ${\bf R}\ni t\to t\chi_{{\bf R}_{+}}(t)$ as
$\ell\to\infty$, and which are uniformly bounded from above by $1+t\chi_{{\bf
R}_{+}}(t)$. Here $\chi_{{\bf R}_{+}}$ is the characteristic function of ${\bf
R}_{+}$. For each $s\in{\bf R}$, let $\psi_{\ell}(z)$ be the solution of the
following auxiliary Monge-Ampère equation

$\displaystyle(\omega+i\partial\bar{\partial}\psi_{\ell})^{n}={\tau_{\ell}(-\varphi-s)\over
A_{\ell,s}}k_{\omega}^{n}(z)\omega_{X}^{n},$ (3.4)

where the constant $A_{\ell,s}$ is defined by

$\displaystyle A_{\ell,s}={1\over
V_{\omega}}\int_{X}\tau_{\ell}(-\varphi-s)k_{\omega}^{n}(z)\omega_{X}^{n}.$
(3.5)

By Yau’s theorem [72], the above equation admits a unique smooth and
$\omega$-plurisubharmonic (PSH) solution $\psi_{\ell}$ normalized by ${\rm
sup}_{X}\psi_{\ell}=0$.

We can now apply Lemma 3, with $a=1$, $b=n/(n+1)$, $q\equiv 0$. Since
$\psi_{\ell}\leq 0$, the condition $-\psi_{\ell}+q\geq 0$ is satisfied, and we
obtain an estimate of the form

$\displaystyle{-\varphi-s\over A_{\ell,s}^{1/(n+1)}}\leq
c_{n,\gamma}(-\psi_{\ell}+c_{n,\gamma}A_{\ell,s})^{n\over n+1}$ (3.6)

where $c_{n,\gamma}$ denote generically positive constants depending only on
$n$ and $\gamma$. Restricting to the set $\Omega_{s}=\\{-\varphi-s>0\\}$, we
can take the $n/(n+1)$-root of both sides, multiply the resulting inequality
by a constant $\beta_{0}$, taking the exponential, and integrate over
$\Omega_{s}$. We obtain

$\displaystyle\int_{\Omega_{s}}{\rm exp}\big{\\{}\beta_{0}({-\varphi-s\over
A_{\ell,s}^{1/(n+1)}})^{n+1\over n}\big{\\}}\omega_{X}^{n}\leq{\rm
exp}(c_{n,\gamma}\beta_{0}A_{\ell,s})\int_{\Omega_{s}}{\rm
exp}(-c_{n,\gamma}\beta_{0}\psi_{\ell})\omega_{X}^{n}.$ (3.7)

We can now invoke the well-known inequality for $\alpha$-invariants, which
states that for all $\omega$ Kähler forms with $\omega\leq\kappa\omega_{X}$
for some fixed $\kappa>0$, there exists a constant $\alpha$ so that,

$\displaystyle\int_{X}{\rm exp}(-\alpha_{0}\psi)\omega_{X}^{n}\leq
C(\alpha_{0},n,\omega_{X},\kappa)$ (3.8)

for any $\alpha_{0}<\alpha$ and any $\omega$-plurisubharmonic function $\psi$
with ${\rm sup}_{X}\psi=0$ [44, 64]. Thus we have

$\displaystyle\int_{\Omega_{s}}{\rm exp}\big{\\{}\beta_{0}({-\varphi-s\over
A_{\ell,s}^{1/(n+1)}})^{n+1\over n}\big{\\}}\omega_{X}^{n}\leq
c_{\alpha_{0},n,\omega_{X},\kappa,\gamma}{\rm
exp}({c_{n,\gamma,\beta_{0}}A_{\ell,s}}).$ (3.9)

We can now let $\ell\to\infty$ and obtain

$\displaystyle\int_{\Omega_{s}}{\rm exp}\big{\\{}\beta_{0}({-\varphi-s\over
A_{s}^{1/(n+1)}})^{n+1\over n}\big{\\}}\omega_{X}^{n}\leq
c_{\alpha_{0},n,\omega_{X},\kappa,\gamma}{\rm
exp}({c_{n,\gamma,\beta_{0}}A_{s}}).$ (3.10)

where $A_{s}={\rm lim}_{\ell\to\infty}A_{\ell,s}$ with $A_{s}$ defined as in
(3.3).

Let $E$ be the energy, defined by

$\displaystyle E={1\over
V_{\omega}}\int_{X}(-\varphi)k_{\omega}^{n}(z)\omega_{X}^{n}.$ (3.11)

Then $A_{s}\leq E$ for any $s\geq 0$, and the preceding inequality implies the
following,

$\displaystyle\int_{\Omega_{s}}{\rm exp}\big{\\{}\beta_{0}({-\varphi-s\over
A_{s}^{1/(n+1)}})^{n+1\over n}\big{\\}}\omega_{X}^{n}\leq c\,{\rm exp}(c\,E)$
(3.12)

which suffices for our purposes.

The inequality (3.10) is the key inequality in our method. It is not difficult
to show that it implies a reverse Hölder inequality. For this, we apply the
Young’s inequality in the following form

$\displaystyle UV\leq U\eta(U)+V\eta^{-1}(V)$ (3.13)

for any monotone strictly increasing function $\eta:{\bf R}_{+}\to{\bf R}_{+}$
with ${\rm lim}_{u\to 0}\eta(u)=0$. Here $\eta^{-1}$ is the inverse of the
function $\eta$. We make the choice $\eta(u)=(\,{\rm log}\,(1+u))^{p}$,
$\eta^{-1}(v)={\rm exp}(V^{1\over p})-1$, and $U=e^{nF_{\omega}}$,
$V=v(z)^{p}$, where we have rewritten $k_{\omega}(z)$ as
$k_{\omega}(z)=c_{\omega}e^{F_{\omega}}$ as in (2.3). This gives

$\displaystyle e^{nF_{\omega}}v(z)^{p}\leq e^{nF_{\omega}}\,{\rm
log}\,^{p}(1+e^{nF_{\omega}})+v(z)^{p}(e^{v(z)}-1)\leq
c_{p}\big{\\{}e^{nF_{\omega}}(1+|nF_{\omega}|^{p})+e^{2v(z)}).$

Next, take

$\displaystyle v(z)={1\over 2}\beta_{0}\Big{(}{-\varphi-s\over
A_{s}^{1/(n+1)}}\Big{)}^{n+1\over n}$ (3.14)

and integrate both sides over $\Omega_{s}$. In view of (3.10), we find

###### Lemma 4

The following inequality holds

$\displaystyle{1\over V_{\omega}}\int_{\Omega_{s}}(-\varphi-s)^{p{n+1\over
n}}k_{\omega}^{n}(z)\omega_{X}^{n}\leq c{c_{\omega}^{n}\over
V_{\omega}}\,A_{s}^{p\over n}(\|e^{nF_{\omega}}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+ce^{cE})$ (3.15)

where $c=C(X,\omega_{X},\beta_{0},p)$ is a constant.

We observe that $A_{s}$ is essentially the $L^{1}$ norm of $-\varphi-s$, so
the inequality we just obtained can be interpreted as a reverse Hölder
inequality. It implies immediately the following growth rate for the function
$\phi(s)$ as in (3.2),

###### Lemma 5

Fix $p>n$. Then we have the inequalities

(a) $A_{s}\leq B_{0}\,\phi(s)^{1+\delta_{0}}$, for $\delta_{0}={(p-n)/np}>0$.

(b) For any $r>0$, $A_{s}\geq r\phi(r+s)$,

(c) $r\phi(s+r)\leq B_{0}\,\phi(s)^{1+\delta_{0}}$,
where the constant $B_{0}=[c{c_{\omega}^{n}\over
V_{\omega}}(\|e^{nF_{\omega}}\|_{L^{1}(\,{\rm log}\,L)^{p}}+ce^{cE})]^{1\over
p}$.

Proof. We begin with the proof of (a). By Hölder’s inequality, we have

$\displaystyle A_{s}\leq\big{\\{}{1\over
V_{\omega}}\int_{\Omega_{s}}(-\varphi-s)^{p{n+1\over
n}}k_{\omega}^{n}(z)\omega_{X}^{n}\big{\\}}^{n\over(n+1)p}({1\over
V_{\omega}}\int_{\Omega_{s}}k_{\omega}^{n}(z)\omega_{X}^{n})^{1\over q}$
(3.16)

where $q$ is defined by ${n\over p(n+1)}+{1\over q}=1$. The first factor on
the right hand side has been shown to be bounded by a multiple of
$A_{s}^{1/(n+1)}$. Putting this factor on the left hand side yields

$\displaystyle A_{s}\leq B_{0}({1\over
V_{\omega}}\int_{\Omega_{s}}e^{nF_{\omega}}\omega_{X}^{n})^{1+n\over
qn}=B_{0}\phi(s)^{1+n\over qn}.$ (3.17)

The exponent $1+n/(qn)$ is readily worked out to be $1+\delta_{0}$,
establishing (a). To see (b), we observe that, trivially,
$\Omega_{r+s}=\\{\varphi<-s-r\\}\subset\Omega_{s}$, and hence

$\displaystyle A_{s}={1\over
V_{\omega}}\int_{\Omega_{s}}(-\varphi-s)e^{nF_{\omega}}\omega_{X}^{n}\geq
r{1\over
V_{\omega}}\int_{\Omega_{s+r}}e^{nF_{\omega}}\omega_{X}^{n}=r\phi(s+r)$ (3.18)

establishing (b). The last statement (c) is a trivial consequence of (a) and
(b). Q.E.D.

We can now invoke a classic lemma of De Giorgi, which says that positive
monotone decreasing functions $\phi(s)$ which tend to $0$ as $s\to\infty$, and
satisfy the growth rate condition stated as (c) in Lemma 5, must vanish for
some $s\geq S_{0}$, where $S_{0}>0$ can be estimated in terms of $B_{0}$ and
$\delta_{0}$. But this implies that $\Omega_{s}$ must be empty for $s\geq
S_{0}$, and hence $\varphi\geq-S_{0}$. Thus we have obtained the following
bound for $\varphi$ [31]:

###### Theorem 1

Let $\varphi$ a solution to the equation where the operator $f(\lambda)$
satisfies the structural conditions listed in (1-4). Assume that the Kähler
form $\omega$ satisfies the condition $\omega\leq\kappa\,\omega_{X}$ for some
positive constant $\kappa$. Fix $p>n$. Then we have the $L^{\infty}$ bound

$\displaystyle\varphi\geq-C$ (3.19)

where $C$ is a constant depending only on $X,\omega_{X},n,p,\gamma,\kappa$ and
the following three quantities

$\displaystyle{c_{\omega}^{n}\over V_{\omega}},\quad E,\quad{\rm
Ent}_{p}(\omega)=\|e^{nF_{\omega}}\|_{L^{1}(\,{\rm log}\,L)^{p}}.$ (3.20)

In the case of the Monge-Ampère equation
$f(\lambda)=(\prod_{j=1})^{n}\lambda_{j})^{1\over n}$, it is easy to see that
${c_{\omega}^{n}\over V_{\omega}}={1\over[\omega_{X}^{n}]}$ and that the
energy $E$ can be bounded by a constant depending only on
$X,\omega_{X},n,\gamma,\kappa$ and ${\rm Ent}_{\\{p=1\\}}(\omega)$. Thus
Theorem 1 gives $L^{\infty}$ estimates for $\varphi$ depending only on the
entropy ${\rm Ent}_{p}(\omega)$ for $p>n$, recovering in this way the
$L^{\infty}$ estimates of Kolodziej [45], even in the more general version
allowing degenerations of the background metric established by Demailly-Pali
[18], and Eyssidieux-Guedj-Zeriahi [23]. We stress however that it holds for
the general class of operators $f(\lambda)$ satisfying the structural
conditions (1-4), which is quite large, as shown by Harvey and Lawson [43].

##  4 Application to energy estimates from entropy

The previous Theorem 1 had reduced $L^{\infty}$ bounds for general non-linear
equations of the form (2.2) to the three quantities
$c_{\omega}^{n}/[\omega^{n}]$, $E$, and ${\rm Ent}_{p}(\omega)$ for $p>n$.
Actually, it has been known for some time that, for fixed background metric
$\omega$, bounds for the energy $E$ can be derived from bounds for the entropy
[3, 9]. This was even one of the key steps in the work of X.X. Chen and J.R.
Cheng [9] on the equation for Kähler metrics of constant scalar curvature. As
shown in [31], the arguments of [9] can be adapted to provide bounds for the
energy, assuming that the entropy is bounded. However, all of these bounds
depend on the background metric $\omega$, and are only useful for fixed
$\omega$.

This problem of bounding uniformly the energy by the entropy is addressed by
the following theorem [37]:

###### Theorem 2

Let $(X,\omega_{X})$ is a compact $n$-dimensional Kähler manifold without
boundary. Let $\omega$ be any Kähler form on $X$ with

$\displaystyle\omega\leq\kappa\,\omega_{X}$ (4.1)

for some constant $\kappa>0$. Consider the equation (2.2) with the operator
$f(\lambda)$ satisfying the conditions (1-4). Then for any $p>0$, any $C^{2}$
solution $\varphi$ to (2.2) satisfies the following

(i) Trudinger-like inequalities

$\int_{X}e^{\alpha(-\varphi)^{q}}\omega_{X}^{n}\leq C_{T},$ (4.2)

(ii) and energy-like estimates

$\int_{X}(-\varphi)^{pq}e^{nF_{\omega}}\omega_{X}^{n}\leq C_{e}.$ (4.3)

Here the exponent $q$ is given by $q=\frac{n}{n-p}$ if $p<n$, and can be any
strictly positive exponent if $p\geq n$. The constants $C_{T}$ and $C_{e}$ are
computable constants depending only on $n,p,q,\omega_{X},\kappa,\gamma$, and
upper bounds for the following two quantities

$\displaystyle{c_{\omega}^{n}\over
V_{\omega}},\quad{\mathrm{Ent}}_{p}({\omega})=\int_{X}e^{nF_{\omega}}|F_{\omega}|^{p}\omega_{X}^{n},$
(4.4)

and the term $\alpha>0$ is a constant that depends only on
$n,p,\gamma,{c_{\omega}^{n}\over V_{\omega}}$ and $\kappa$.

We observe that, in the case of a fixed background Kähler metric $\omega$,
this theorem was proved as Theorem 3 in [31]. As stressed above, the point of
the new theorem is to have uniform estimates, even as the background metric
$\omega$ may degenerate to the boundary of the Kähler cone. For this same
reason, we consider the case $p>n$. When $p>n$, when the background Kähler
form $\omega$ is fixed, it follows from [31], Theorem 1, that the solution
$\varphi$ of the equation is actually bounded, and the above Trudinger-like
and energy-like estimates follow at once. But here again, the existing results
do not give the estimates uniform in $\omega$ that we seek.

We only describe the version of the auxiliary complex Monge-Ampère equation
that we need, together with the comparison inequality, leaving fuller details
to [38].

Thus let $a=pq=\frac{np}{n-p}$ ($a$ is any positive number if $p=n$). We solve
the following complex Monge-Ampère equation

$(\omega+i\partial\bar{\partial}\psi_{s,\ell})^{n}=\frac{\tau_{\ell}(-\varphi-s)^{a}}{A_{s,\ell}}c_{\omega}^{n}e^{nF_{\omega}}\omega_{X}^{n},\quad{\rm
sup}_{X}\psi_{s,\ell}=0.$ (4.5)

Here the constant $A_{s,\ell}$ is defined by

$A_{s,\ell}=\frac{c^{n}}{V_{\omega}}\int_{X}\tau_{\ell}(-\varphi-s)^{a}e^{nF_{\omega}}\omega_{X}^{n}$
(4.6)

to make the equation (4.5) compatible. By assumption $[\omega]$ is a Kähler
class, so by Yau’s theorem [72], (4.5) admits a unique smooth solution
$\psi_{s,\ell}$. We observe that as $\ell\to\infty$

$A_{s,\ell}\to
A_{s}:=\frac{c_{\omega}^{n}}{V_{\omega}}\int_{\Omega_{s}}(-\varphi-s)^{a}e^{nF_{\omega}}\omega_{X}^{n}.$
(4.7)

Next we consider comparison functions of the form

$\Phi:=-\varepsilon(-\psi_{s,\ell}+\Lambda)^{b}-\varphi-s,$

where $\varepsilon$ and $\Lambda$ are constants, and $b$ is a suitable power.
Using the identities in Section 2 and the maximum principle, we then show that

$\displaystyle\Phi\leq 0$ (4.8)

if the constants are given by

$b=\frac{n}{n+a}\in(0,1),\quad\mbox{and
}\varepsilon=\frac{1}{\gamma^{1/(n+a)}(nb)^{n/(n+a)}}A_{s,\ell}^{\frac{1}{n+a}},$
(4.9)

and $\Lambda$ is chosen so that $\varepsilon b\Lambda^{-(1-b)}=1$, that is,

$\Lambda=\frac{b^{1/(1-b)}}{(\gamma^{1/(n+a)}(nb)^{n/(n+a)})^{1/(1-b)}}A_{s,\ell}^{\frac{1}{a}}.$
(4.10)

The inequality $\Phi\leq 0$ on $X$ implies that $-\varphi-s$ can be controlled
by $c_{a,n,\gamma}A_{s,\ell}^{1\over n+a}(-\psi_{s,\ell}+A_{s,\ell}^{1\over
a})^{b}$, after which the theorem can be established following the template of
Section 3.

##  5 Application to stability estimates

It is not difficult to adapt the same method of $L^{\infty}$ estimates to
stability estimates. As in Section 3, let $(X,\omega_{X})$ be a compact Kähler
manifold, and $\omega$ be a Kähler metric such that
$\omega\leq\kappa\omega_{X}$ for some $\kappa>0$. We consider the following
complex Monge-Ampère equations

$(\omega+i\partial\bar{\partial}u)^{n}=c_{\omega}e^{f}\omega_{X}^{n},\mbox{
and }(\omega+i\partial\bar{\partial}v)^{n}=c_{\omega}e^{h}\omega_{X}^{n},$
(5.1)

with the constant $c_{\omega}=\int_{X}\omega^{n}$ and
$\int_{X}e^{f}\omega_{X}^{n}=\int_{X}e^{h}\omega_{X}^{n}=1$. The functions
$u,v$ are normalized such that

${\rm max}_{X}(u-v)={\rm max}_{X}(v-u).$

Assume that for some $K>0$ and $p>n$,

$\max(\|e^{f}\|_{L^{1}(\,{\rm
log}\,L)^{p}(X,\omega_{X})},\|e^{h}\|_{L^{1}(\,{\rm
log}\,L)^{p}(X,\omega_{X})})\leq K.$

###### Theorem 3

Under these assumptions, there is a constant $C>0$ depending on
$n,p,\omega_{X},\kappa$, and $K$ such that

${\rm sup}_{X}|u-v|\leq C\|e^{f}-e^{h}\|_{L^{1}}^{\beta},$

where $\beta=\beta(n,p)=(n+3+\frac{p-n}{pn})^{-1}>0$.

We outline the main idea of the proof, and refer to [33] for the details. Fix
a small $r>0$. First we may assume $\int_{\\{v\leq
u\\}}(e^{f}+e^{h})\omega_{X}^{n}\leq 1$. We then consider the auxiliary
equation

$\displaystyle(\omega+i\partial\bar{\partial}\psi)^{n}=c_{\omega}{\tau_{\ell}(-\varphi+q(z)-s)\over
A_{\ell,s}}e^{h}\omega_{X}^{n},\quad{\rm sup}_{X}\psi=0,$ (5.2)

with $\varphi=v-(1-r)u$, $q(z)=-3\beta_{0}r$, where $\beta_{0}$ is an upper
bound of $\|u\|_{L^{\infty}}$ and $\|v\|_{L^{\infty}}$. $\beta_{0}$ exists
from Section 3. Take the comparison function

$\displaystyle\Phi=-\varepsilon(-\psi+\Lambda)^{n\over n+1}-\varphi+q(z)-s$
(5.3)

with $\varepsilon=c_{1}(n)A_{\ell,s}^{1/(n+1)}$ and
$\Lambda=c_{2}(n)(\varepsilon/r)^{n+1}$ for appropriate constants $c_{1}(n)$
and $c_{2}(n)$. Arguing as in Section 3, we arrive at $v-u\geq-Cr$ for a
constant $C>0$ depending only on $n,p,\kappa,\omega_{X}$ and $K>0$.

We remark that analogous stability estimates as in Theorem 3 hold for complex
Hessian equations as well [33, 21].

##  6 Application to the nef case

So far, we have considered equations whose background form $\omega$ is Kähler
(although it may degenerate to the boundary of the Kähler cone). In this
section, following [34], we show how it can be adapted to nef classes.

As above, $(X,\omega_{X})$ is a Kähler manifold. We assume only the class
$[\omega]$ is Kähler and $\omega\leq\kappa\omega_{X}$, but the $(1,1)$-form
$\omega$ may not be positive.

We consider both the Monge-Ampère and Hessian equations. While $L^{\infty}$
estimates do not hold in the usual form, the following estimates can be
established [34]:

###### Theorem 4

Assume that $(X,\omega_{X})$ is an $n$-dimensional compact Kähler manifold,
and let $\omega$ be a closed $(1,1)$-form which may not be positive, but
$[\omega]$ is a Kähler class with $\omega\leq\kappa\,\omega_{X}$.

(a) Let $f(\lambda)=(\prod_{j=1}^{n}\lambda_{j})^{1\over n}$, corresponding to
the Monge-Ampère equation. Recall that the envelope associated to $\omega$ is
the $\omega$-PSH function defined by

$\displaystyle{\mathcal{V}}_{\omega}=\sup\\{u|~{}u\in PSH(X,\omega),\,u\leq
0\\}.$ (6.1)

Then for any fixed $p>n$, there is a constant $C$ depending only on
$\omega_{X},n,p,\kappa,{\rm Ent}_{p}(\omega)$, so that the solution $\varphi$
of the equation (2.2) satisfies

$\displaystyle 0\leq-\varphi+{\mathcal{V}}_{\omega}\leq C.$ (6.2)

(b) Let $f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$, corresponding to the
$k$-th Hessian equation. Define the envelope ${\mathcal{V}}_{\omega,k}$
corresponding to the $\Gamma_{k}$ cone by

$\displaystyle{\mathcal{V}}_{\omega,k}={\rm sup}\,\\{v\leq 0\\}$ (6.3)

where $v$ runs over non-negative $C^{2}$ functions with the vector of
eigenvalues of the relative endomorphism
$\omega_{X}^{-1}(\omega+i\partial\bar{\partial}v)$ lying in the $\Gamma_{k}$
cone. Define the energy $E_{k}$ by

$\displaystyle
E_{k}=\int_{X}(-\varphi+{\mathcal{V}}_{\omega,k})e^{kF_{\omega}}\omega_{X}^{n}.$
(6.4)

Recall also the constant $c_{\omega}$ as defined for the equation (2.2). Then
for any $p>n$, there is a constant $C$ depending only on
$\omega_{X},n,o,\kappa$, $E_{k}$, and $\|e^{{k}F_{\omega}}\|_{L^{1}(\,{\rm
log}\,L)^{p}}$, ${c_{\omega}^{n}\over[\omega]^{k}[\omega_{X}]^{n-k}}$ so that

$\displaystyle 0\leq-\varphi+{\mathcal{V}}_{\omega,k}\leq C.$ (6.5)

We note that Part (a) on the Monge-Ampère equation had been proved with
different methods by Boucksom et al [6] and Fu-Guo-Song [24]. Part (b) on
Hessian equations is new and due to [34].

As in earlier applications, we indicate only the auxiliary Monge-Ampère
equation to be used, and the comparison function $\Phi$. We only discuss the
proof of Part (a), the one of Part (b) being similar.

A first technical difficulty is that ${\mathcal{V}}_{\omega}$ is only
$C^{1,1}$. However, it has been shown by Berman [3] that there exists a
sequence of smooth and strictly $\omega$-PSH functions
$\\{u_{\beta}\\}_{\beta=1}^{\infty}$ which converge uniformly to
${\mathcal{V}}_{\omega}$. In the following we can use $u_{\beta}$ in place of
${\mathcal{V}}_{\omega}$, and then take limit $\beta\to\infty$.

The auxiliary Monge-Ampère equation is then

$\displaystyle(\omega+i\partial\bar{\partial}\psi)^{n}={\tau_{\ell}(-\varphi+u_{\beta}-s)\over
A_{\ell,\beta}}k_{\omega}^{n}(z)\omega_{X}^{n},\quad{\rm sup}_{X}\psi=-1$
(6.6)

and the comparison function $\Phi$ is defined by

$\displaystyle\Phi=-\varepsilon(-\psi+u_{\beta}+1+\Lambda)^{n\over
n+1}-\varphi+u_{\beta}-s.$ (6.7)

In the general setting of Section 2, this corresponds to our general ansatz
with $q=\tilde{q}=u_{\beta}$ and the simple shift in the normalization of
$\psi$, with ${\rm sup}_{X}\psi=-1$. We note that an analogue of (2.19) still
holds by the fact that $\omega_{u_{\beta}}>0$ although $\omega$ may not be
positive. We can then show that $\Phi\leq 0$, and establish the desired bounds
following our general template.

##  7 Application to the modulus of continuity

It has been shown by Kolodziej [45] that the solution of the complex Monge-
Ampère equation on a compact Kähler manifold is Hölder continuous if the right
hand side is of class $L^{q}$ for some $q>1$. He has also shown that if the
solution is in some Orlicz space, then it must be continuous. But even so, his
arguments are not direct, and we don’t have any information on the modulus of
continuity of the solution. The modulus of continuity is typically a delicate
question. However, it turns out that it can also be addressed using the class
of auxiliary Monge-Ampère equations discussed in the present paper.

More specifically, let $(X,\omega_{X})$ be again a compact Kähler manifold of
complex dimension $n$. We consider the complex Monge-Ampère equation with
$\int_{X}e^{F}\omega_{X}^{n}=\int_{X}\omega_{X}^{n}$

$(\omega_{X}+i\partial\bar{\partial}\varphi)^{n}=e^{F}\omega_{X}^{n},\quad\omega_{\varphi}=\omega_{X}+i\partial\bar{\partial}\varphi>0.$
(7.1)

Then the following estimate is established in [35]:

###### Theorem 5

Fix $p>n$. Then we have

$|\varphi(x)-\varphi(y)|\leq\frac{C}{|\,{\rm
log}\,d(x,y)|^{\alpha}},\quad\forall x,y\in X$

for some constant $C>0$ depending on $n,p,\omega_{X}$ and
$\|e^{F}\|_{L^{1}(\,{\rm log}\,L)^{p}}$. Here $d(x,y)$ denotes the geodesic
distance of $x,y$ in the Riemannian manifold $(X,\omega_{X})$, and
$\alpha=\min\\{\frac{p}{n+1},\frac{p-n}{n}\\}>0$.

The proof of Theorem 5 relies on the following auxiliary complex Monge-Ampère
equation:

$\displaystyle(\omega_{X}+i\partial\bar{\partial}\psi)^{n}={\tau_{\ell}(-\varphi+q(z)-s)\over
A_{\ell,s}}e^{F}\omega_{X}^{n}$ (7.2)

with $q=(1-|\,{\rm log}\,\delta|^{-p/(n+1)})\varphi_{\delta}-2\delta$ and
$\varphi_{\delta}$ is the (rescaled) Kiselman-Legendre transform of $\varphi$
at level $\delta>0$. With the comparison function

$\displaystyle\Phi=-\varepsilon(-\psi+\Lambda)^{n\over n+1}-\varphi+q(z)-s,$
(7.3)

we can argue as in Section 3 to conclude that $\varphi_{\delta}-\varphi\leq
2\delta+|\,{\rm log}\,\delta|^{-p/(n+1)}\varphi_{\delta}+S_{\infty}$ for
$S_{\infty}={C}{|\,{\rm log}\,\delta|^{-(p-n)/n}}$. The proof of Theorem 5
then follows from the fact that $\varphi_{\delta}(z)$ is equal to
$\max\varphi$ over the ball $B(z,\delta)$ up to a controlled error term.

##  8 Application to the Hermitian case

A striking feature of the above method of auxiliary Monge-Ampère equations is
the ease with which it can be adapted to the case of Hermitian manifolds. Thus
we let $(X,\omega_{X})$ be a compact Hermitian manifold with $\omega_{X}$ a
fixed Hermitian metric,
$\omega_{\varphi}=\omega_{X}+i\partial\bar{\partial}\varphi$,
$h_{\varphi}=\omega^{-1}_{X}\omega_{\varphi}$, and consider the equation

$\displaystyle
f(\lambda[h_{\varphi}])=e^{F},\qquad\lambda[h_{\varphi}]\in\Gamma,\qquad{\rm
sup}_{X}\varphi=0,$ (8.1)

where the operator $f(\lambda)$ defined on a cone $\Gamma$ satisfies the
structural conditions (1-4) spelled out in section §2. We have then [37]

###### Theorem 6

For any $p>n$, and $C^{2}$ solution $\varphi$ of the equation (8.1) must
satisfy the $L^{\infty}$ bound

$\displaystyle{\rm sup}_{X}|\varphi|\leq C$ (8.2)

where $C$ is a constant depending only on $X,\omega_{X},n,p,\gamma$ and
$\|e^{nF}\|_{L^{1}(\,{\rm log}\,L)^{p}}$.

In the case of the Monge-Ampère equation, $L^{\infty}$ estimates were first
obtained in the Hermitian setting by Cherrier [14] and Tosatti and Weinkove
[65], assuming a pointwise bound for $e^{F}$. The sharper version with entropy
bounds $\|e^{nF}\|_{L^{1}(\,{\rm log}\,L)^{p}}$ was obtained by Dinew and
Kolodziej [20], and required a highly non-trivial extension of pluripotential
theory to the Hermitian setting. An approach based on envelopes has been
recently proposed by Guedj and Lu [30]. Our theorem applies to much more
general classes of equations, and its proof is arguably the simplest, as it
bypasses the complicated integration by parts with torsion terms which arise
in Hermitian geometry. We provide a brief sketch.

A first observation is that the previous auxiliary complex Monge-Ampère
equation cannot be applied as it is. The reason is that, on a compact
Hermitian manifold, unlike on Kähler manifolds, solutions exist only up to an
undetermined constant. Because of this, we shall use instead as auxiliary
equation the Dirichlet problem for a complex Monge-Ampère equation on a
Euclidean ball, which has been shown by Caffarelli, Kohn, Nirenberg, and
Spruck [7] to admit always a smooth solution.

Thus, fix $r_{0}$ small enough, but depending only on $(X,\omega_{X})$ so
that, for any $z_{0}\in X$, there is a coordinate system $z$ centered at
$z_{0}$ so that

$\displaystyle{1\over 2}i\partial\bar{\partial}|z|^{2}\leq\omega_{X}\leq
2i\partial\bar{\partial}|z|^{2}\mbox{ in }B(z_{0},2r_{0})=\\{|z|<2r_{0}\\}.$
(8.3)

Let $x_{0}\in X$ be a point where $\varphi$ attains its minimum. We shall show
that

$\displaystyle\varphi(x_{0})\geq-C$ (8.4)

where $C$ is a constant depending only on
$X,\omega_{X},p,\gamma,\|e^{nF}\|_{L^{1}(\,{\rm log}\,L)^{p}}$, and
$\|\varphi\|_{L^{1}(X,\omega_{X})}$. It is not difficult to show by a separate
argument that $\|\varphi\|_{L^{1}(X,\omega_{X})}$ is bounded by a constant
depending only $n,\omega_{X}$, for all functions with ${\sup}_{X}\varphi=0$
with
$\lambda[h_{\varphi}]\in\Gamma\subset\Gamma_{1}=\\{\lambda;\lambda_{1}+\cdots+\lambda_{n}>0\\}$.
The desired theorem would follow.

We introduce now the auxiliary equation. Let $\Omega=B(x_{0},2r_{0})$, and for
each $s$ with $0<s<2r_{0}^{2}$, set

$\displaystyle u_{s}(z)=\varphi(z)-\varphi(x_{0})+{1\over 2}|z|^{2}-s$ (8.5)

and $\Omega_{s}=\\{z\in\Omega;u_{s}(z)<0\\}$. Let $\psi_{s,\ell}$ be the
solution of the following Dirichlet problem,

$\displaystyle(i\partial\bar{\partial}\psi_{s,\ell})^{n}={\tau_{\ell}(u_{s})\over
A_{s,\ell}}e^{nF(z)}\omega_{X}^{n}\quad{\rm on}\ \Omega,\quad\psi_{s,\ell}=0\
{\rm on}\ \partial\Omega,$ (8.6)

where the coefficients $A_{s,\ell}$ are defined by

$\displaystyle
A_{s,\ell}=\int_{\Omega}\tau_{\ell}(-u_{s})e^{nF}\omega_{X}^{n}\ \to\
A_{s}=\int_{\Omega_{s}}(-u_{s})e^{nF}\omega_{X}^{n},\mbox{ as }\ell\to\infty.$
(8.7)

By [7], the solution $\psi_{s,\ell}$ of this Dirichlet problem exists and is
unique.

We can now state the key comparison inequality between $u_{s}$ and
$\psi_{s,\ell}$

$\displaystyle-u_{s}\leq C(n,\gamma)\,A_{s,\ell}^{1\over
n+1}(-\psi_{s,\ell})^{n\over n+1}$ (8.8)

on $\bar{\Omega}$, where $C(n,\gamma)$ depends only on $n$ and $\gamma$. Note
that the auxiliary Monge-Ampère equation is of the general form considered in
(2.6), with $\omega=0$, $\tilde{q}(z)=0$, $\Lambda=0$, and
$q(z)=\varphi(z_{0})-{1\over 2}|z|^{2}$, (and $s\to-s$). In particular
$q_{\bar{k}j}=-{1\over 2}\delta_{kj}$. It is then easy to apply the maximum
principle to the function

$\displaystyle\Phi=-\varepsilon(-\psi_{s,\ell})^{n\over n+1}-u_{s}$ (8.9)

with the preliminary computations as in Lemma 2, and the desired inequality
follows with $\varepsilon^{n+1}=A_{s,\ell}\gamma^{-1}(n+1)^{n}n^{-2n}$.

We can now follow the template provided by the compact Kähler case. First we
establish the inequality

$\displaystyle\int_{\Omega_{s}}{\rm exp}\\{\beta{(-u_{s})^{n+1\over n}\over
A_{s}^{1/n}}\\}\omega_{X}^{n}\leq C$ (8.10)

where $\beta$ and $C$ are strictly positive constants depending only on
$n,\gamma,r_{0}$. This is the analogue in the Hermitian case of the inequality
(3.12). For this, we apply Young’s inequality as in the compact Kähler case,
with the $\alpha$-invariant replaced by the following inequality of Kolodziej
[45] for plurisubharmonic functions $\psi$

$\displaystyle\int_{D}e^{-\alpha\psi}dV\leq C$ (8.11)

on bounded pseudoconvex domains $D\subset{\bf C}^{n}$ with $\psi=0$ on
$\partial D$, and Monge-Ampère measure
$\int_{D}(i\partial\bar{\partial}\psi)^{n}=1$. Here $\alpha$ is a strictly
positive constant111Kolodziej’s inequality is a generalization to all
dimensions of a classic inequality in one complex dimension due to Brezis and
Merle [5]. Kolodziej’s original proof used pluripotential theory. A more
recent PDE proof has been provided by Wang, Wang, and Zhou [69]..

Next, we can apply Young’s inequality to the exponential inequality (8.10) and
obtain for any $p>n$ the following reverse Hölder inequality

$\displaystyle A_{s}\leq
C_{0}(\int_{\Omega_{s}}e^{nF}\omega_{X}^{n})^{1+\delta_{0}}$ (8.12)

with $\delta_{0}={1\over n}-{1\over p}$, and $C$ is a constant depending only
on $\omega_{X},n,p,\gamma$ and $\|e^{nF}\|_{L^{1}(\,{\rm log}\,L)^{p}}$.
Setting

$\displaystyle\phi(s)=\int_{\Omega_{s}}e^{nF}\omega_{X}^{n}$ (8.13)

we can rewrite the reverse inequality as $A_{s}\leq
C_{0}(\phi(s))^{1+\delta_{0}}$. And since it is easily seen that $A_{s}\geq
t\int_{\Omega_{s-t}}e^{nF}\omega_{X}^{n}$, we conclude that for any
$t\in(0,s)$

$\displaystyle t\phi(s-t)\leq C_{0}(\phi(s))^{1+\delta_{0}}.$ (8.14)

We can now apply a version of De Giorgi’s lemma for monotone increasing and
positive functions $\phi(s)$ on $(0,s_{0})$ satisfying ${\rm lim}_{s\to
0^{+}}\phi(s)=0$ and the growth rate (8.14), which says that under these
conditions, there must exist a constant $c_{0}>0$ depending only on
$s_{0},C_{0}$ and $\delta_{0}$ so that

$\displaystyle\phi(s_{0})\geq c_{0}.$ (8.15)

Finally, we note the elementary inequality

$\displaystyle\phi(s_{0})\,{\rm log}\,(-\varphi(x_{0})-s_{0})^{1\over
2}\leq\int_{\Omega_{s_{0}}}\,{\rm
log}\,{-\varphi-2^{-1}|z|^{2}\over(-\varphi(x_{0})-s_{0})^{1\over
2}}e^{nF}\omega_{X}^{n}$ (8.16)

which results itself from the elementary inequality $-\varphi-{1\over
2}|z|^{2}>-\varphi(x_{0})-s_{0}$ on $\Omega_{0}$ since we may assume without
loss of generality that $s_{0}<{1\over 2}$ and $\varphi(x_{0})<-2$. Applying
the Young’s inequality to the integrand on the right hand side, we find

$\displaystyle\phi(s_{0})\,{\rm log}\,(-\varphi(x_{0})-s_{0})^{1\over
2}\leq{1\over 2}{\rm max}\big{\\{}\|e^{nF}\|_{L^{1}(\,{\rm
log}\,L)^{p}}+C^{\prime}_{p}C_{1}(\omega_{X}),C_{p}^{\prime\prime}{\|\varphi\|_{L^{1}}+C_{2}(\omega)\over(-\varphi(x_{0})-s_{0})^{1\over
2}}\big{\\}}$

Since $\phi(s_{0})$ is bounded from below by a constant, it follows that
$-\varphi(x_{0})-s_{0}$ must be bounded from above. The theorem is proved.

##  9 Application to $(n-1)$-form Monge-Ampère equations

A new generation of problems in complex geometry, notably from non-Kähler
geometry and mathematical physics, has led to non-linear equations of the form
$f(\lambda)=e^{F}$, but where $\lambda$ are not the eigenvalues of a hessian
matrix $h_{\varphi}=\omega_{X}^{-1}(\omega+i\partial\bar{\partial}\varphi)$,
but of a more general matrix involving the derivatives of $\varphi$ to second
order. One basic example of such an equation is the so-called $(n-1)$-form
Monge-Ampère equation, solved by G. Székelyhidi, V. Tosatti, and B. Weinkove
[63] in their solution of the Gauduchon conjecture. Central to the study of
this equation is the $L^{\infty}$ estimate. In this section, we show how our
method of auxiliary Monge-Ampère equations can be used to establish this
estimate, with in fact slightly weaker and more general hypotheses than in
[63].

Indeed, let $(X,\omega_{X})$ be a compact Hermitian manifold. If $\omega$ is
any other Hermitian metric, we consider the smooth functions $\varphi$ for
which the form

$\displaystyle\tilde{\omega}_{\varphi}=\omega+{1\over
n-1}(\Delta_{\omega_{X}}\varphi\omega_{X}-i\partial\bar{\partial}\varphi)$
(9.1)

is positive, where
$\Delta_{\omega_{X}}\varphi=n{i\partial\bar{\partial}\varphi\wedge\omega_{X}^{n-1}\over\omega_{X}^{n}}$
is the rough Laplacian with respect to the metric $\omega_{X}$. Then

###### Theorem 7

Let $\tilde{h}_{\varphi}=\omega_{X}^{-1}\tilde{\omega}_{\varphi}$, and
$\lambda[\tilde{h}_{\varphi}]$ be its eigenvalues. Fix any $p>n$. Then any
smooth solution $\varphi$ of the equation

$\displaystyle\prod_{j=1}^{n}(\tilde{\lambda}_{\varphi})_{j}=e^{nF}$ (9.2)

with $\tilde{\omega}_{\varphi}>0$ and ${\rm sup}_{X}\varphi=0$ satisfies

$\displaystyle{\rm sup}_{X}|\varphi|\leq C$ (9.3)

where $C$ is a constant depending only on $\omega_{X},n,p,\omega$ and
$\|e^{nF}\|_{L^{1}(\,{\rm log}\,L)^{p}(X,\omega_{X})}$.

We observe that this result is more precise than in [67, 66] in the sense that
only the norm $\|e^{nF}\|_{L^{1}(\,{\rm log}\,L)^{p}(X,\omega_{X})}$ is
needed, and not pointwise norms. In [62], G. Székelyhidi obtains an estimate
which depends on $\|e^{nF}\|_{L^{q}}$ for $q>2$, using Blocki’s approach [3]
based on the Alexandrov-Bakelman-Pucci (ABP) maximum principle.

We sketch the proof. A first observation is that the tensor
$\Theta^{i\bar{j}}$, as defined in [67] by

$\displaystyle\Theta^{k\bar{j}}={1\over n-1}(({\rm
Tr}_{\tilde{\omega}_{\varphi}}\omega_{X})\omega_{X}^{k\bar{j}}-\tilde{\omega}_{\varphi}^{k\bar{j}})$
(9.4)

is positive definite. In fact, in coordinates where
$(\omega_{X})_{j\bar{k}}=\delta_{jk}$,
$(\tilde{\omega}_{\varphi})_{j\bar{k}}=\lambda_{j}\delta_{jk}$ at a given
point, then $\Theta^{k\bar{j}}={1\over
n-1}(\sum_{\ell\not=j}{1\over\lambda_{\ell}})\delta_{jk}$.

Next, fix $r_{0}$ small enough so that, at any point $z_{0}\in X$, there is a
holomorphic coordinate system $z$ with

$\displaystyle{1\over 2}i\partial\bar{\partial}|z|^{2}\leq\omega_{X}\leq
2i\partial\bar{\partial}|z|^{2},\qquad z\in B(z_{0},2r_{0})=\\{|z|<2r_{0}\\}.$
(9.5)

Let $x_{0}$ be a minimum point for $\varphi$, $\Omega=B(x_{0},2r_{0})$, and
fix a small constant $\varepsilon^{\prime}>0$ depending only on
$X,\omega_{X},\omega$ so that

$\displaystyle\omega\geq{2\varepsilon^{\prime}\over n-1}({\rm
Tr}_{\omega_{X}}\omega)\omega_{X}.$ (9.6)

Define now for $s\in(0,s_{0})$, $s_{0}=4\varepsilon^{\prime}r_{0}^{2}$,

$\displaystyle
u_{s}(z)=\varphi(z)-\varphi(x_{0})+\varepsilon^{\prime}|z|^{2}-s,\qquad
z\in\Omega.$ (9.7)

Then $u_{s}(z)>0$ for $z\in\partial\Omega$, and hence the sublevel set
$\Omega_{s}=\\{z|u_{s}(z)<0\\}\cap\Omega$ is relatively compact in $\Omega$
and also an open set. We can now consider the solution $\psi_{s,\ell}$ of the
following auxiliary Dirichlet problem on $\Omega$ for the complex Monge-Ampère
equation,

$\displaystyle(i\partial\bar{\partial}\psi_{s,\ell})^{n}={\tau_{\ell}(-u_{s})\over
A_{s,\ell}}e^{nF}\omega_{X}^{n}\quad{\rm on}\ \Omega,\quad\psi_{s,\ell}=0\
{\rm on}\ \partial\Omega$ (9.8)

with $A_{s,\ell}$ defined by

$\displaystyle
A_{s,\ell}=\int_{\Omega}\tau_{\ell}(-u_{s})e^{nF}\omega_{X}^{n}\to
A_{s}=\int_{\Omega_{s}}(-u_{s})e^{nF}\omega_{X}^{n}.$ (9.9)

By [7], this Dirichlet problem admits a unique solution $\psi_{s,\ell}$. From
the very equation, the solution $\psi_{s,\ell}$ satisfies
$\int_{\Omega}(i\partial\bar{\partial}\psi_{s,\ell})^{n}=1$.

The key comparison estimate is now

$\displaystyle-u_{s}\leq({n+1\over n})^{n\over n+1}A_{s,\ell}^{1\over
n+1}(-\psi_{s,\ell})^{n\over n+1}$ (9.10)

which follows from the non-positivity on $\Omega$ of the test function

$\displaystyle\Phi=-\varepsilon(-\psi_{s,\ell})^{n\over
n+1}-u_{s},\qquad\varepsilon=A_{s,\ell}^{1\over n+1}({n+1\over n})^{n\over
n+1}.$ (9.11)

To show that $\Phi\leq 0$, we apply the maximum principle at a maximum point
$x_{0}$ for $\Psi$, but with the operator $Lv=\Theta^{j\bar{k}}v_{\bar{k}j}$,
so that $L\Phi(x_{0})\leq 0$. The desired inequality follows using the
arithmetic-geometric inequality, along the same lines as Lemmas.

Once we have the key comparison estimate (9.10), the proof follows the same
template as in the compact Kähler case, or more precisely the compact
Hermitian case. Thus we apply the exponential estimate of Kolodziej to arrive
at a reverse Hölder inequality, which allows an application of De Giorgi’s
lemma, and the proof is completed using a general $L^{1}$ bound for
$-\varphi$. This completes the proof of Theorem 7.

Finally, we note that there has been considerable interest recently in more
general equations involving $f(\lambda[\tilde{h}_{\varphi}])$, motivated in
part by non-Kähler geometry, mirror symmetry, and equations from string
theories (see e.g. [26, 16, 56, 50]). In particular, subsolutions and general
$C^{2}$ estimates have been obtained in [28, 62]. A frequent feature of these
new equations is the appearance of gradient terms. For related developments,
see [25, 29, 15, 51, 52, 53] and references therein.

##  10 Application to Green’s functions

Perhaps surprisingly, the auxiliary Monge-Ampère equations which we have been
discussing turn out to very effective in many seemingly unrelated problems in
geometric analysis. In this section, we show how they can be applied to derive
lower bounds for the Green’s function. The Green’s function is the solution of
a linear partial differential equation, so this would be a case of a linear
problem solved by a non-linear method. Such methods had been instrumental in
the study of Schrödinger equations [13, 57, 49, 59], and in the celebrated
work of M. Kuranishi [46, 47, 48] on the embeddability of strongly pseudo-
convex manifolds, as pointed out by C. Fefferman. However, it does not appear
that comparisons with the Monge-Ampère equation had been used before in linear
problems. In our case, this non-linear method is needed because it gives
bounds which are uniform in the underlying geometry.

In this section, we describe some of the results obtained in [36]. The setting
is then a compact Kähler manifold $(X,\omega_{X})$ of dimension $n$, and a
closed nonnegative $(1,1)$-form $\chi$ whose cohomology class $[\chi]$ is nef
and big, that is, $[\chi]$ lies in the closure of the Kähler cone and
$\int_{X}\chi^{n}>0$. Then $[\chi+t\omega_{X}]$ is a Kähler class for any
$t\in(0,1]$. For any $\varepsilon>0$, $N>0$, $\gamma\geq 1$, and any fixed
$t\in(0,1]$, we introduce the following class of Kähler metrics,

$\displaystyle{\cal
M}_{t}^{\prime}(N,\varepsilon,\gamma)=\\{\omega\in[\chi+t\omega_{X}];{1\over
V}\int_{X}e^{(1+\varepsilon)F_{\omega}}\omega_{X}^{n}\leq N,\ {\rm
sup}_{X}e^{-F_{\omega}}\leq\gamma^{-1}\\}.$ (10.1)

where $F_{\omega}$ denotes the relative volume form of $\omega$,

$\displaystyle F_{\omega}=\,{\rm
log}\,({\omega^{n}/[\omega^{n}]\over\omega_{X}^{n}/[\omega_{X}^{n}]}).$ (10.2)

We are interested in bounds which hold uniformly for $\omega\in{\cal
M}^{\prime}_{t}(N,\varepsilon,\gamma)$. The following is a basic example of an
estimate which would be well-known and readily established for a fixed Kähler
form $\omega$, but requires a non-linear method in order to have uniform
bounds with respect to $\omega\in{\cal M}_{t}^{\prime}(N,\varepsilon,\gamma)$:

###### Theorem 8

Let $\omega$ be any Kähler form in the class ${\cal
M}_{t}^{\prime}(N,\varepsilon,\gamma)$.

(a) Let $v\in L^{1}(X,\omega^{n})$ be any function satisfying
$\int_{X}v\omega^{n}=0$. Let $\Omega_{s}=\\{v>s\\}$ be the super-level sets of
$v$. Assume that

$\displaystyle v\in C^{2}(\bar{\Omega}_{0}),\qquad\Delta_{\omega_{t}}v\geq-a\
\ {\rm in}\ \Omega_{0}$ (10.3)

for some $a>0$. Then

$\displaystyle{\rm sup}_{X}v\leq C(a+\|v\|_{L^{1}(X,\omega^{n})})$ (10.4)

where $C$ is a constant depending only on $n,\omega_{X},\chi,\varepsilon,N$,
and $\gamma$.

(b) Assume now that $v\in C^{2}(X)$ and that

$\displaystyle|\Delta_{\omega}v|\leq 1,\quad{\rm
and}\quad\int_{X}v\omega^{n}=0.$ (10.5)

Then there is a constant $C$ depending only on
$n,\omega_{X},\chi,\varepsilon,N,\gamma$ such that
$\|v\|_{L^{1}(X,\omega^{n})}\leq C$.

The proof relies again on an auxiliary complex Monge-Ampère equation.
Replacing $v$ by $v/a$ we may assume $a=1$. Thus let as before
$u\to\tau_{\ell}(u)$ be again a sequence of smooth strictly positive functions
which decrease to the function $u\to u\chi_{{\bf R}_{+}}(u)$ as
$\ell\to\infty$, and consider the solution $\psi_{s,\ell}$ of the following
equation

$\displaystyle(\omega+i\partial\bar{\partial}\psi_{s,\ell})^{n}={\tau_{\ell}(v-s)\over
A_{s,\ell}}\omega^{n},\qquad{\rm sup}_{X}\psi_{s,\ell}=0,$ (10.6)

where $A_{s,\ell}$ is again a normalizing constant

$\displaystyle
A_{s,\ell}={1\over[\omega^{n}]}\int_{X}\tau_{\ell}(v-s)\omega^{n}\ \to\
{1\over[\omega^{n}]}\int_{\Omega_{s}}(v-s)\omega^{n}=A_{s}$ (10.7)

as $\ell\to\infty$. Yau’s theorem insures then the existence of a unique
solution $\psi_{s,\ell}$.

We need to compare $v$ to $\psi_{s,\ell}$. For this, we express $\omega$ by
the $\partial\bar{\partial}$-lemma as

$\displaystyle\omega=\chi+t\omega_{X}+i\partial\bar{\partial}\varphi,\qquad{\rm
sup}_{X}\varphi=0$ (10.8)

for a unique function $\varphi$. Then the key estimate is the following: there
exists a constant $\Lambda$ depending only on $n,p,\chi,\omega_{X},{\rm
Ent}_{p}(\omega)$ so that

$\displaystyle-\psi+\varphi+\Lambda\geq 1,\quad{\rm and}\quad\Phi\leq 0$
(10.9)

and the function $\Phi$ is defined by

$\displaystyle\Phi=-\varepsilon(-\psi+\varphi+\Lambda)^{n\over n+1}+v-s$
(10.10)

with $\varepsilon^{n+1}=({n+1\over n^{2}})^{n}(a+\varepsilon
n)^{n}A_{s,\ell}$. The proof of this is analogous to the ones we have used for
the previous auxiliary complex Monge-Ampère equations.

Once we know that $\Phi\leq 0$, we can apply an $\alpha$-invariant inequality,
uniform for all Kähler classes bounded by a fixed multiple of $\omega_{X}$ to
obtain an inequality of the form

$\displaystyle\int_{\Omega_{s}}{\rm exp}(\alpha{(v-s)^{n+1\over n}\over
A_{s,\ell}^{1/n}})\omega_{X}^{n}\leq C$ (10.11)

and hence, using Young’s inequality, the following reverse Hölder inequality
for $p>n$,

$\displaystyle\int_{\Omega_{s}}(v-s)^{(n+1)p\over n}e^{F}\omega_{X}^{n}\leq
C\,A_{s,\ell}^{p\over n}\to C\,A_{s}^{p\over n}.$ (10.12)

This readily implies
$A_{s}\leq(\int_{\Omega_{s}}e^{F}\omega_{X}^{n})^{1+n\over np^{\prime}}$, with
$p^{\prime}$ the dual exponent of $p(n+1)/n$. An easy consequence is the
growth inequality

$\displaystyle r\phi(s+r)\leq C\phi(s)^{1+\delta_{0}},\qquad s\geq 0,\ r>0$
(10.13)

for the monotone decreasing function
$\phi(s)=\int_{\Omega_{s}}e^{F}\omega_{X}^{n}$. Again by a De Giorgi lemma, it
follows that $\phi(s)$ must vanish for $s>S_{0}$, where $S_{0}$ can be
estimated by the constants in the growth condition. Thus $v\leq S_{0}$, and
part (a) of Theorem 8 is proved.

Next, we sketch the proof of Part (b) of Theorem 8. We observe that the proof
of Part (a) did not require a uniform lower bound $\gamma$ for the volume form
in the definition of the class ${\cal M}^{\prime}_{t}(N,\varepsilon,\gamma)$,
but the proof of Part (b) will.

We argue by contradiction. Thus assume that there exists a sequence of metrics
$\omega_{j}\in{\cal M}_{t_{j}}^{\prime}(N,\varepsilon,\gamma)$ with
$\\{t_{j}\\}_{j}\subset(0,1)$ and a sequence of functions $\hat{v}_{j}\in
C^{2}(X)$ satisfying $\|\hat{v}_{j}\|_{L^{1}(X,\omega_{j}^{n})}=1$ and

$\displaystyle\Delta_{\omega_{j}}\hat{v}_{j}=\hat{h}_{j},\quad\int_{X}\hat{v}_{j}\omega_{j}^{n}=0,\quad{\rm
sup}_{X}|\hat{h}_{j}|\to 0$ (10.14)

as $j\to\infty$. Multiplying the above equation by $\hat{v}_{j}$ and
integrating by parts gives

$\displaystyle\int_{X}|{\nabla}\hat{v}_{j}|_{\omega_{j}}^{2}\omega_{j}^{n}=|\int_{X}\hat{h}_{j}\hat{v}_{j}\omega_{j}^{n}|\leq
V_{\omega_{j}}^{1\over 2}\,{\rm sup}_{X}|\hat{h}_{j}|\to 0.$ (10.15)

On the other hand, we can write

$\displaystyle\int_{X}|{\nabla}\hat{v}_{j}|_{\omega_{X}}\omega_{X}^{n}$
$\displaystyle\leq$
$\displaystyle\int_{X}(|{\nabla}\hat{v}_{j}|_{\omega_{j}}^{2}{\rm
tr}_{\omega_{X}}\omega_{j})^{1\over 2}\omega_{X}^{n}$ (10.16)
$\displaystyle\leq$
$\displaystyle(\int_{X}|{\nabla}\hat{v}_{j}|_{\omega_{j}}^{2}e^{F_{j}}\omega_{X}^{n})^{1\over
2}(\int_{X}({\rm tr}_{\omega_{X}}\omega_{j})e^{-F_{j}}\omega_{X}^{n})^{1\over
2}.$

The first factor in the right hand side tends to $0$ as $j\to\infty$, since

$\displaystyle\int_{X}|{\nabla}\hat{v}_{j}|_{\omega_{j}}^{2}e^{F_{j}}\omega_{X}^{n}={V_{X}\over
V_{\omega_{j}}}\,\int_{X}|{\nabla}\hat{v}_{j}|_{\omega_{j}}^{2}\omega_{j}^{n}\to
0$ (10.17)

since $V_{\omega_{j}}\geq[\chi^{n}]$ for all $\omega_{j}$. In view of the
lower bound for the volume form of metrics $\omega_{j}\in{\cal
M}_{t_{j}}^{\prime}(N,\varepsilon,\gamma)$, we have

$\displaystyle\int_{X}({\rm
tr}_{\omega_{X}}\omega_{j})e^{-F_{j}}\omega_{X}^{n}\leq{1\over
n\gamma}\int_{X}\omega_{j}\wedge\omega_{X}^{n-1}\leq C$ (10.18)

where $C$ is a constant independent of $j$. This implies that

$\displaystyle\int_{X}|{\nabla}\hat{v}_{j}|_{\omega_{X}}\omega_{X}^{n}\to
0\quad{\rm as}\ j\to\infty.$ (10.19)

From this, it is not difficult to deduce that $\hat{v}_{j}$ is uniformly
bounded in the Sobolev space $W^{1,1}(X,\omega_{X})$, and that, ater passing
to subsequences, it must converge to a constant $\hat{v}_{\infty}$ in
$L^{1}(X,\omega_{X}^{n})$. This constant can on one hand be verified to be
nonzero in view of the normalization
$\|\hat{v}_{j}\|_{L^{1}(X,\omega_{j}^{n})}=1$, and on the other hand to be
$0$, in view of the condition $\int_{X}\hat{v}_{j}\omega_{j}^{n}=0$ satisfied
by $\hat{v}_{j}$. This is a contradiction and Theorem 8 is proved.

We can now establish uniform estimates for the Green’s function. Recall now
that the Green’s function with respect to the Kähler metric $\omega$ is
defined by the equations

$\displaystyle\Delta_{\omega}G(x,y)=-\delta_{x}(y)+{1\over
V_{\omega}},\qquad\int_{X}G(x,y)\omega^{n}=0.$ (10.20)

###### Theorem 9

Fix $\varepsilon>0,N>0$ and $\gamma\in(0,1)$. Then for any Kähler metric
$\omega\in{\cal M}_{t}^{\prime}(\varepsilon,N,\gamma)$, the corresponding
Green’s function $G(x,y)$ satisfies the following estimates, with constants
$C$ which depend only on $n,\omega_{X},\chi,\varepsilon,N$ and $\gamma$:

(a) $\int_{X}|G(x,\cdot)|\omega^{n}\leq C$;

(b) ${\rm inf}_{y\in X}G(x,y)\geq-C$;

(c) For any $\delta>0$, there is a constant $C_{\delta}$ depending
additionally on $\delta$ so that, for all $x\in X$,

$\displaystyle\int_{X}|G(x,\cdot)|^{{n\over
n-1}-\delta}\omega^{n}+\int_{X}|{\nabla}G(x,\cdot)|^{{2n\over
2n-1}-\delta}\omega^{n}\leq C_{\delta}.$ (10.21)

Parts (a) and (b) of Theorem 9 are direct consequences of Theorem 8. For
example, for any fixed $x$, the function $v=-G(x,y)$ is in
$C^{\infty}(X\setminus\\{x\\})$ and satisfies the conditions in Part (a) of
Theorem 8 with $\Delta_{\omega}v(y)=-{1\over
V_{\omega}}\geq-{1\over[\chi^{n}]}$ for $y\in\\{v\geq 0\\}$. Thus Theorem 8
applies, giving the lower bound

$\displaystyle{\rm inf}_{y\in
X}G(x,y)\geq-C(1+\|G(x,\cdot)\|_{L^{1}(X,\omega^{n})})$ (10.22)

for some constant depending only on $n,p,\omega_{X},\chi,N$. With a bit more
work, we can deduce from Part (b) of Theorem 8 that
$\|G(x,\cdot)\|_{L^{1}(X,\omega^{n})}\leq C$. Combined with the preceding
inequality, we obtain Parts (a) and (b) of Theorem 9.

The proof of Part (c) is harder and requires a new idea, involving comparisons
with another auxiliary complex Monge-Ampère equation. The key inequality to be
established is a uniform bound for the $L^{q}$ norm of $G(x,\cdot)$,

$\displaystyle\int_{X}|G(x,y)|^{q}\omega^{n}(y)\leq C_{q}$ (10.23)

first for $q\in 1+{1\over r_{0}}$ for some $r_{0}>n$, and then iteratively for
any $q<{n\over n-1}$. By Part (b), we can add a uniform constant to
$G(x,\cdot)$ to obtain a function ${\cal G}(x,\cdot)\geq 1$. Fix $r_{0}>n$ and
a large $k>>1$ and set

$\displaystyle H_{k}(y)={\rm min}\\{{\cal G}(x,y),k\\}$ (10.24)

which we assume is smooth, by smoothing it out if necessary. The above
integral is closely related to the following integral

$\displaystyle\int_{X}{\cal G}(x,y)H_{k}(y)^{1\over r_{0}}\omega^{n}(y)$
(10.25)

which is itself closely related to the solution $u_{k}$ of the following
Laplace equation

$\displaystyle\Delta_{\omega}u_{k}=-H_{k}^{1\over r_{0}}+{1\over
V_{\omega}}\int_{X}H_{k}^{1\over r_{0}}\omega^{n},\qquad{1\over
V_{\omega}}\int_{X}u_{k}\omega^{n}=0.$ (10.26)

To estimate $u_{k}$, we introduce another auxiliary complex Monge-Ampère
equation,

$\displaystyle(\omega+i\partial\bar{\partial}\psi_{k})^{n}={H_{k}^{n\over
r_{0}}+1\over V_{\omega}^{-1}\int_{X}(H_{k}^{n\over
r_{0}}+1)\omega^{n}}\omega^{n},\qquad{\rm sup}_{X}\psi_{k}=0.$ (10.27)

It can then be shown that

$\displaystyle{\rm sup}_{X}|\psi_{k}|\leq C$ (10.28)

and that

$\displaystyle\varepsilon^{\prime}u_{k}+(\psi_{k}-\varphi)-{1\over
V_{\omega}}\int_{X}(\psi_{k}-\varphi)\omega^{n}\leq C$ (10.29)

where $C$ and $\varepsilon^{\prime}$ are uniform constants, and $\varphi$ is
the potential for the Kähler metric $\omega\in[\chi+t\omega_{X}]$ introduced
earlier in (10.8). From here, the desired inequality follows.

Finally, we establish integral bounds for ${\nabla}G(x,y)$. First we note the
elementary inequality

$\displaystyle\int_{X}{|{\nabla}_{y}{\cal G}(x,y)|_{\omega(y)}^{2}\over{\cal
G}(x,y)^{1+\beta}}\omega^{n}(y)\leq{1\over\beta}$ (10.30)

which holds for all $\beta>0$, and follows from applying Green’s formula to
$u(y)={\cal G}(x,y)^{-\beta}$. Next, setting

$\displaystyle H_{k}(y)={\rm min}\\{{|{\nabla}_{y}{\cal
G}(x,y)|_{\omega(y)}^{2}\over{\cal G}(x,y)^{1+\beta}},k\\}$ (10.31)

and arguing as in the estimate of $\|G(x,\cdot)\|_{L^{1+{1\over
r_{0}}}(X,\omega^{n})}$, we can show that

$\displaystyle\int_{X}{\cal G}(x,y)H_{k}(y)^{1\over r_{0}}\omega^{n}(y)\leq
C.$ (10.32)

The desired $L^{s}(X,\omega^{n})$ bound for ${\nabla}G(x,\cdot)$ ultimately
follows. Q.E.D.

It is instructive to compare the preceding theorem with the classic result of
Cheng and Li [12] on lower bounds for the Green’s function in Riemannian
geometry. This result asserts the existence of a uniform lower bound depending
only the dimension, the diameter, the volume, and a lower bound on the Ricci
curvature. In our Kähler setting, Theorem 9 is easily seen to imply the
following theorem, where only a lower bound on the scalar curvature, combined
with an integral estimate for the volume form, suffice to give lower bounds
for the Green’s function which are uniform in $\omega$:

###### Theorem 10

Let $\omega$ be any Kähler metric in $[\omega_{X}]$. Then if
$\|e^{F_{\omega}}\|_{L^{1+\varepsilon}(X,\omega_{X})}\leq N$ for some
$\varepsilon>0,N>0$, and the scalar curvature $R(\omega)$ satisfies
$R(\omega)\geq-\kappa$ for some $\kappa\geq 0$, then the Green’s function of
$(X,\omega)$ satisfies

$\displaystyle{\rm inf}_{y\in X}G(x,y)\geq-C,\quad x\in X,$ (10.33)

for a constant $C$ depending only on $n,\omega_{X},\varepsilon,N$, and
$\kappa$.

We note that the class ${\cal M}_{t}^{\prime}(\varepsilon,N,\gamma)$ of
Hermitian metrics which we have used so far is not the only class to which the
methods of this section apply. Other classes are described in [36] as well,
with a key difference being the replacement of a pointwise lower bound on
$e^{F_{\omega}}$ by an integral bound. In fact, for applications to diameter
bounds to be described in the next section, it will be important to relax
further the constant lower bound $\gamma$ to a non-negative function which may
vanish along a closed set of Hausdorff dimension strictly less than $2n-1$.

In the remaining part of this section, we describe some applications of the
above bounds for the Green’s function to a priori estimates for the complex
Monge-Ampère equation. A priori estimates are often obtained by applying the
maximum principle to elliptic differential inequalities satisfied by the
quantity under consideration. Sharp lower bounds for the Green’s function can
provide a more effective tool, especially if we consider conditions involving
integrals. The following is a sample of sharp estimates which can be obtained
in this manner.

Recall that we have assumed that the $(1,1)$-form $\chi$ is non-negative, and
the Kähler class $[\chi]$ is big. By Kodaira’s lemma, there is an effective
divisor $D$ in $X$ such that

$\displaystyle\chi-\varepsilon_{0}{\rm Ric}(h_{D})\geq\delta_{0}\,\omega_{X}$
(10.34)

for some positive constants $\varepsilon_{0},\delta_{0}$, and a Hermitian
metric $h_{D}$ on the line bundle $[D]$ associated with $D$. Let $s_{D}$ be a
holomorphic section defining $D$ with ${\rm sup}_{X}|s_{D}|_{h_{D}}^{2}=1$.
Let $\omega$ be any metric in $[\chi+t\omega_{X}]$. Let $\varphi$ be its
potential, i.e. $\omega=\chi+t\omega_{X}+i\partial\bar{\partial}\varphi$ and
${\rm sup}_{X}\varphi=0$. Thus, in the notation (10.2) for the relative volume
form $F_{\omega}$, $\varphi$ satisfies the complex Monge-Ampère equation

$\displaystyle(\chi+t\omega_{X}+i\partial\bar{\partial}\varphi)^{n}=ce^{F_{\omega}}\omega_{X}^{n},\qquad
c={V_{\omega}\over V_{X}},\quad{\rm sup}_{X}\varphi=0.$ (10.35)

###### Theorem 11

Fix $\varepsilon,N,\gamma\in(0,1)$ and $p>n$. Then for any $t\in(0,1]$ and
$\omega\in{\cal M}_{t}^{\prime}$, we have the estimate

$\displaystyle|{\nabla}\varphi|_{\omega_{X}}^{2}\leq{C\over|s_{D}|_{h_{D}}^{2A}}$
(10.36)

where $C$ depends only on $n,\varepsilon,\chi,\omega_{X},N,\gamma,p$, and
$A>0$ depends only on $n,\varepsilon,\chi,\omega_{X},N,\gamma$.

Note that gradient bounds had been obtained before, but under pointwise
assumptions on $|{\nabla}F|$ [4, 55]. An earlier result requiring an $L^{p}$
bound for $|{\nabla}F|$ with $p\geq 2n$ is in [11, 32]. The range $p>n$ in the
above theorem is sharp.

###### Theorem 12

Under the same assumptions as in Theorem 11, but with $p>2n$, we have the
estimate

$\displaystyle|i\partial\bar{\partial}\varphi|_{\omega_{X}}^{2}\leq{C\over|s_{D}|_{h_{D}}^{2B}}$
(10.37)

where $C$ depends only on $n,\varepsilon,\chi,\omega_{X},N,\gamma,p$ and
$\int_{X}|{\nabla}F_{\omega}|_{\omega_{X}}^{p}e^{F_{\omega}}\omega_{X}^{n}$,
and $B>0$ depends only on $n,\varepsilon,\chi,\omega_{X},N,\gamma$.

We have formulated the above estimates for families of degenerating metrics.
But even in the case of a fixed background Kähler form $\omega$, the above
estimates improve on the known ones. For example, we have

###### Theorem 13

Consider the complex Monge-Ampère equation

$\displaystyle(\omega_{X}+i\partial\bar{\partial}\varphi)^{n}=e^{F}\omega_{X}^{n},\quad{\rm
sup}_{X}\varphi=0$ (10.38)

on an $n$-dimensional compact Kähler manifold $(X,\omega_{X})$. Assume that
$F$ satisfies the condition

$\displaystyle\|e^{F}\|_{L^{1+\varepsilon}(X,\omega_{X})}\leq N$
$\displaystyle{\rm sup}_{X}e^{-F}\leq\gamma^{-1},$ (10.39)

Then for any $p>2n$, we have

(a) ${\rm sup}_{X}|i\partial\bar{\partial}\varphi|_{\omega_{X}}^{2}\leq C$,
where $C>0$ is a constant depending only on
$n,p,\omega_{X},\varepsilon,N,\gamma$ and
$\int_{X}|{\nabla}F|_{\omega_{X}}^{p}e^{F}\omega_{X}^{n}$.

(b) ${\rm
sup}_{X}|{\nabla}i\partial\bar{\partial}\varphi|_{\omega_{X}}^{2}\leq C$,
where $C>0$ is a constant depending only on
$n,p,\omega_{X},\varepsilon,N,\gamma$,

$\displaystyle\gamma,\int_{X}|{\nabla}F|_{\omega_{X}}^{p}e^{F}\omega_{X}^{n},\quad\int_{X}|i\partial\bar{\partial}F|_{\omega_{X}}^{p}e^{F}\omega_{X}^{n},$
(10.40)

and upper and lower bounds for the endomorphism
$\omega_{X}^{-1}(\omega_{X}+i\partial\bar{\partial}\varphi)$.

For the estimate in (a) to hold in general, we do need $p\geq 2n$. We also
note that previous $C^{3}$ bounds had required a $C^{3}$ bound for $F$, and
that the proof of (b) also relied on the approach of [54], which relied on
estimating the connection forms instead of the potentials.

##  11 Application to diameter bounds

In general, estimates for the Green’s function can imply estimates for the
diameter of the underlying metric. This can be seen as follows [39].

Let $x_{0},y_{0}$ be points with $d_{\omega}(x_{0},y_{0})={\rm
diam}(X,\omega)$, and define the function $d$ on $X$ by
$d(y)=d_{\omega}(x_{0},y)$. Then $d$ is a Lipschitz function with Lipschitz
constant $1$. The Green’s formula applied to $d(y)$ gives

$\displaystyle
d(x)={1\over[\omega^{n}]}\int_{X}d(y)\omega(y)^{n}+\int_{X}\langle{\nabla}_{y}G(x,y),{\nabla}d(y)\rangle_{\omega(y)}\omega(y)^{n}.$
(11.1)

The fact that $d(x_{0})=0$ gives

$\displaystyle{1\over[\omega^{n}]}\int_{X}d(y)\omega(y)^{n}=-\int_{X}\langle{\nabla}_{y}G(x_{0},y),{\nabla}d(y)\rangle_{\omega(y)}\omega(y)^{n}\leq\int_{X}|{\nabla}_{y}G(x_{0},y)|_{\omega(y)}\omega(y)^{n}.$

We can then write

$\displaystyle{\rm diam}(X,\omega)=d(y_{0})$ $\displaystyle=$
$\displaystyle{1\over[\omega^{n}]}\int_{X}d(y)\omega(y)^{n}+\int_{X}\langle{\nabla}_{y}G(y_{0},y),{\nabla}d(y)\rangle_{\omega(y)}\omega(y)^{n}$
$\displaystyle\leq$
$\displaystyle\int_{X}|{\nabla}_{y}G(x_{0},y)|_{\omega(y)}\omega(y)^{n}+\int_{X}|{\nabla}_{y}G(y_{0},y)|_{\omega(y)}\omega(y)^{n}$

which shows that the diameter can be estimated by an integral bound for the
gradient of the Green’s function.

Thus the bounds obtained in the previous section already imply some diameter
bounds. However, for many geometric applications, such as diameters in the
Kähler-Ricci flow, it is important to relax the conditions on the lower bound
$\gamma$ for the volume form. It turns out that this is possible, albeit quite
non-trivial. Thus the following theorems were established in [39]:

###### Theorem 14

Let $(X,\omega_{X})$ be an $n$-dimensional connected compact Kähler manifold.
For given parameters $A,K>0,p>n$ and $\gamma$ a continuous non-negative
function, let ${\cal V}(X,\omega_{X},n,$ $A,p,K,\gamma)$ be the following
space of Kähler metrics

$\displaystyle{\cal
V}(X,\omega_{X},n,A,p,K,\gamma)=\\{\omega;[\omega]\cdot[\omega_{X}]^{n-1}\leq
A,\,{\cal N}_{X,\omega_{X},p}(\omega)\leq
K,\,{\omega^{n}\over\omega_{X}^{n}}\geq\gamma V_{\omega}\\}$ (11.2)

where ${\cal N}_{X,\omega_{X},p}(\omega)$ is the $p$-Nash entropy, defined by

$\displaystyle{\cal
N}_{X,\omega_{X},p}(\omega)=\int_{X}|F|^{p}e^{F}\omega_{X}^{n}=\|e^{F}\|_{L^{1}(\,{\rm
log}\,L)^{p}(\omega_{X})},\quad F={1\over
V_{\omega}}{\omega^{n}\over\omega_{X}^{n}}.$ (11.3)

Assume that

$\displaystyle{\rm dim}_{\cal H}\\{\gamma=0\\}<2n-1$ (11.4)

where ${\rm dim}_{\cal H}$ denotes the Hausdorff dimension. Then for any
$A,K>0$ and $p>n$, there exist constants $C,c>0$ depending only on
$X,\omega_{X},n,A,p,K,\gamma$ and $\alpha$ depending only on $n$ and $p$ so
that

(a)
$\int_{X}|G(x,\cdot)|\omega^{n}+\int_{X}|{\nabla}G(x,\cdot)|\omega^{n}+(-{\rm
inf}_{y\in X}G(x,y))V_{\omega}\leq C$;

(b) ${\rm diam}\,(X,\omega)\leq C$;

(c) ${Vol_{\omega}(B_{\omega}(x,R))\over Vol_{\omega}(X)}\geq c\,R^{\alpha}$
for any $x\in X$, $R\in(0,1)$.

We stress that the above theorem can give bounds on the diameter even when no
lower bound on the Ricci curvature is available. This is of particular
importance for the Kähler-Ricci flow. More generally, we obtain the following
Kähler analogue of Gromov’s precompactness theorem for metric spaces:

###### Theorem 15

Let $(X,\omega_{X})$ be a connected $n$-dimensional Kähler manifold, and let
$\gamma$ be a non-negative function with

$\displaystyle{\rm dim}_{\cal H}\\{\gamma=0\\}<2n-1.$ (11.5)

Then for any $A,K>0$, $p>n$, any sequence $\\{\omega_{j}\\}$ in ${\cal
V}(X,\omega_{X},n,A,p,K,\gamma)$ admits a subsequence converging in Gromov-
Hausdorff topology to a compact metric space $(X_{\infty},d_{\infty})$.

Several applications of these theorems to the Kähler-Ricci flow and to the
asymptotic behavior of fibrations near the singular fibre can be found in
[39].

##  12 Application to the taming of symplectic forms

Let $(X,J)$ be a compact almost complex manifold with $J$ the almost complex
structure. Suppose $m=2n$ is the real dimension of $X$. A Riemannian metric
$\tilde{g}$ on $X$ is called almost Kähler, if $\tilde{g}$ is $J$-compatible,
i.e. $\tilde{g}(JY,JZ)=\tilde{g}(Y,Z)$ for any vector fields $Y,Z$ and the
associated $2$-form $\omega_{\tilde{g}}$ defined by
$\omega_{\tilde{g}}(Y,Z)=\tilde{g}(JY,Z)$ is closed.

Let $\Omega$ be a taming symplectic form, that is, $\Omega(Y,JY)>0$ for
$Y\not=0$, and $g$ be the associated almost Hermitian metric of $\Omega$, i.e.

$g(Y,Z)=\frac{1}{2}\Omega(Y,JZ)+\frac{1}{2}\Omega(Z,JY),\quad\forall\mbox{
vector fields }Y,Z.$

Write $dV_{g}$ for the volume form of the Riemannian metric $g$. For a smooth
function $F\in C^{\infty}(X)$ normalized by
$\int_{X}e^{F}dV_{g}=\int_{X}\Omega^{n}$, we consider the following Calabi-Yau
equation on $X$

${\rm det}\tilde{g}=e^{2F}{\rm det}g,$ (12.1)

where we require that $\omega_{\tilde{g}}$ is an almost Kähler form with
$[\omega_{\tilde{g}}]=[\Omega]$. As shown by Donaldson [22], the existence of
solutions to this equation would have important consequences in symplectic
geometry.

It has been proved by Tosatti, Weinkove, and Yau in [68] that the $C^{2}$-a
priori estimates of $\tilde{g}$ satisfying the equation (12.1) can be derived
by the $L^{\infty}$ estimates of $\varphi\in C^{\infty}(X)$, which satisfies
the linear equation

$\Delta_{\tilde{g}}\varphi=2n-2n\frac{\omega_{\tilde{g}}^{n-1}\wedge\Omega}{\omega_{\tilde{g}}^{n}}=2n-{\rm
tr}_{\tilde{g}}g,\quad{\rm sup}_{X}\varphi=0.$ (12.2)

Here $\Delta_{\tilde{g}}$ is the usual Riemannian Laplacian operator of the
metric $\tilde{g}$. Note that the equation (12.2) admits a unique solution
since the function on the right-hand-side has integral zero against the volume
form $dV_{\tilde{g}}$.

We assume

$\|e^{F}\|_{L^{2}(X,dV_{g})}^{2}=\int_{X}e^{2F}dV_{g}\leq K$

for some constant $K>0$. Our main result is the following $L^{\infty}$
estimate of $\varphi$ in (12.2):

###### Theorem 16

Suppose $\tilde{g}$ is an almost Kähler metric solving the equation (12.1) and
$\varphi$ solves (12.2). Then there exists a constant $C>0$ depending on $n,g$
and $K$ such that

${\rm sup}_{X}|\varphi|\leq C(1+\|\varphi\|_{L^{1}(X,e^{2F}dV_{g})}).$

In the following, we denote $\\{x^{1},\ldots,x^{m}\\}$ a local real
coordinates on some open subset of $X$. Let $\tilde{g}$ be an almost Kähler
metric, and $\tilde{\omega}(Y,Z)=\tilde{g}(JY,Z)$ be the associated symplectic
$2$-form. Locally we have

$\tilde{g}=\tilde{g}_{ij}dx^{i}\otimes
dx^{j},\quad\tilde{\omega}=\frac{1}{2}\tilde{\omega}_{ij}dx^{i}\wedge
dx^{j},\quad J=J_{i}^{j}dx^{i}\otimes\frac{\partial}{\partial x^{j}},$ (12.3)

where the summations are taken over $i,j\in\\{1,2,\cdots,m\\}$. It follows
from straightforward calculations that

$\tilde{\omega}_{ij}=\tilde{g}_{ik}J_{j}^{k},\quad\tilde{\omega}_{ij}=-\tilde{\omega}_{ji}.$
(12.4)

$\tilde{\omega}$ being almost Kähler means

$0=d\tilde{\omega}=\frac{1}{2}\frac{\partial\tilde{\omega}_{ij}}{\partial
x^{l}}dx^{l}\wedge dx^{i}\wedge
dx^{j}=\frac{1}{6}(\frac{\partial\tilde{\omega}_{ij}}{\partial
x^{l}}+\frac{\partial\tilde{\omega}_{li}}{\partial
x^{j}}+\frac{\partial\tilde{\omega}_{jl}}{\partial x^{i}})dx^{l}\wedge
dx^{i}\wedge dx^{j}$

in other words,

$\frac{\partial\tilde{\omega}_{ij}}{\partial
x^{l}}+\frac{\partial\tilde{\omega}_{li}}{\partial
x^{j}}+\frac{\partial\tilde{\omega}_{jl}}{\partial x^{i}}=0,\quad\forall
i,j,l.$ (12.5)

Multiplying both sides of (12.5) by
$\tilde{\omega}^{ij}:=\tilde{g}^{ik}J_{k}^{j}$ which is skew-symmetric in
$i,j$, and taking summation over $i,j$, we get

$\tilde{g}^{ik}J_{k}^{j}\frac{\partial\tilde{\omega}_{ij}}{\partial
x^{l}}+2\tilde{g}^{ik}J_{k}^{j}\frac{\partial\tilde{\omega}_{jl}}{\partial
x^{i}}=0.$ (12.6)

Substituting (12.4) to (12.6), we obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\tilde{g}^{ik}J_{k}^{j}\frac{\partial\tilde{g}_{ip}}{\partial
x^{l}}J_{j}^{p}+\tilde{g}^{ik}J_{k}^{j}\tilde{g}_{ip}\frac{\partial
J_{j}^{p}}{\partial
x^{l}}+2\tilde{g}^{ik}J_{k}^{j}\frac{\partial\tilde{g}_{jp}}{\partial
x^{i}}J_{l}^{p}+2\tilde{g}^{ik}J_{k}^{j}\tilde{g}_{jp}\frac{\partial
J_{l}^{p}}{\partial x^{i}}$ $\displaystyle=$
$\displaystyle-\tilde{g}^{ik}\frac{\partial\tilde{g}_{ik}}{\partial
x^{l}}+J_{k}^{j}\frac{\partial J_{j}^{k}}{\partial
x^{l}}+2\tilde{g}^{ik}J_{k}^{j}\tilde{g}_{jp}\frac{\partial
J_{l}^{p}}{\partial
x^{i}}+2\tilde{g}^{ik}\frac{\partial\tilde{g}_{kl}}{\partial x^{i}}$
$\displaystyle-2\tilde{g}^{ik}\tilde{g}_{jp}J_{l}^{p}\frac{\partial
J_{k}^{j}}{\partial
x^{i}}-2\tilde{g}^{ik}\tilde{g}_{jp}J_{k}^{j}\frac{\partial
J_{l}^{p}}{\partial x^{i}}$ $\displaystyle=$
$\displaystyle-\tilde{g}^{ik}\frac{\partial\tilde{g}_{ik}}{\partial
x^{l}}+2\tilde{g}^{ik}\frac{\partial\tilde{g}_{kl}}{\partial
x^{i}}+J_{k}^{j}\frac{\partial J_{j}^{k}}{\partial
x^{l}}-2J_{p}^{i}\frac{\partial J_{l}^{p}}{\partial
x^{i}}+2\tilde{g}^{ik}\tilde{g}_{lp}J_{j}^{p}\frac{\partial
J_{k}^{j}}{\partial x^{i}}+2J_{p}^{i}\frac{\partial J_{l}^{p}}{\partial
x^{i}},$ $\displaystyle=$
$\displaystyle-\tilde{g}^{ik}\frac{\partial\tilde{g}_{ik}}{\partial
x^{l}}+2\tilde{g}^{ik}\frac{\partial\tilde{g}_{kl}}{\partial
x^{i}}+J_{k}^{j}\frac{\partial J_{j}^{k}}{\partial
x^{l}}+2\tilde{g}^{ik}\tilde{g}_{lp}J_{j}^{p}\frac{\partial
J_{k}^{j}}{\partial x^{i}},$

from which we derive that

$\tilde{g}^{ik}\frac{\partial\tilde{g}_{kl}}{\partial
x^{i}}-\frac{1}{2}\tilde{g}^{ik}\frac{\partial\tilde{g}_{ik}}{\partial
x^{l}}=-\frac{1}{2}J_{k}^{j}\frac{\partial J_{j}^{k}}{\partial
x^{l}}-\tilde{g}^{ik}\tilde{g}_{lp}J_{j}^{p}\frac{\partial J_{k}^{j}}{\partial
x^{i}}.$ (12.7)

It then follows that

$\tilde{g}^{ik}\tilde{\Gamma}_{ik}^{q}=\tilde{g}^{ql}(\tilde{g}^{ik}\frac{\partial\tilde{g}_{kl}}{\partial
x^{i}}-\frac{1}{2}\tilde{g}^{ik}\frac{\partial\tilde{g}_{ik}}{\partial
x^{l}})=-\frac{1}{2}\tilde{g}^{ql}J_{k}^{j}\frac{\partial J_{j}^{k}}{\partial
x^{l}}-\tilde{g}^{ik}J_{j}^{q}\frac{\partial J_{k}^{j}}{\partial x^{i}}.$
(12.8)

From (12.8), we see that the second term in the Laplacian of a function $u$,
$\Delta_{\tilde{g}}u=\tilde{g}^{ij}\frac{\partial^{2}u}{\partial x^{i}\partial
x^{j}}-\tilde{g}^{ik}\tilde{\Gamma}_{ik}^{q}\frac{\partial u}{\partial x^{q}}$
is independent of the first order derivatives of the metric coefficients
$\tilde{g}_{ij}$.

Let $x_{0}\in X$ be a minimum point of $\varphi$, i.e.
$\varphi(x_{0})=\inf_{X}\varphi$. Choose a fixed number $r_{0}>0$ such that
$2r_{0}\leq$ the injectivity radius of the fixed Riemannian manifold $(X,g)$.
Take the normal coordinates of $(X,g)$ centered at $x_{0},$
$(U,\\{x^{1},\ldots,x^{m}\\})$. Without loss of generality we may assume that
on $U$ the following holds

$\frac{1}{2}\delta_{ij}\leq g_{ij}\leq 2\delta_{ij},$ (12.9)

and the Euclidean ball

$B(x_{0},2r_{0})=\\{x\in U:|x|<2r_{0}\\}\subset\subset U$

where $|x|=\sqrt{\sum_{j=1}^{m}(x^{j})^{2}}$ is the usual Euclidean norm of
the coordinates $x=(x^{1},\ldots,x^{m})$. We also have a constant
$C_{J}^{\prime}>0$ depending on $J,g$ such that

${\rm sup}_{U}\Big{(}|\sum_{j,k}J_{k}^{j}\frac{\partial J_{j}^{k}}{\partial
x^{l}}|_{g}+|\sum_{j}J_{j}^{q}\frac{\partial J_{k}^{j}}{\partial
x^{i}}|_{g}\Big{)}\leq C_{J}^{\prime}.$ (12.10)

From the equation (12.8), we have for any smooth function $\psi$ on $U$, there
exists a uniform constant $C_{J}>0$ such that on $U$

$|\tilde{g}^{ik}\tilde{\Gamma}^{q}_{ik}\psi_{q}|\leq
C_{J}|\nabla\psi|_{g}\cdot{\rm tr}_{\tilde{g}}g,$ (12.11)

where $\psi_{q}=\frac{\partial\psi}{\partial x^{q}}$ and
$|\nabla\psi|_{g}^{2}=g^{ij}\psi_{i}\psi_{j}$ is the gradient of $\psi$ with
respect to the fixed metric $g$. We emphasize that the constant $C_{J}$ in
(12.11) depends only on $g,J$ and can be made to be independent of the choice
of coordinates, though the LHS of (12.11) is only locally defined. Indeed, we
can see from (12.10) that $C_{J}^{\prime}$ depends on $|J|_{g}$ and
$|\nabla_{g}J|_{g}$, both of which are globally defined.

Let $\eta\in(0,1)$ be a small positive constant to be determined. For any
$0<s\leq s_{0}=\eta r_{0}^{2}$, we consider the function defined on
$B(x_{0},2r_{0})$

$u_{s}(x):=\varphi(x)-\varphi(x_{0})+\eta|x|^{2}-s.$ (12.12)

By the choice of the point $x_{0}$, it is clear that $u_{s}\geq-s$. Define the
sublevel set of $u_{s}$ by

$\Omega_{s}:=\\{x\in B(x_{0},2r_{0})|~{}u_{s}(x)<0\\}.$ (12.13)

Note that $x_{0}\in\Omega_{s}$ so $\Omega_{s}$ is a nonempty open subset of
$B(x_{0},2r_{0})$. Moreover, by the choice of $s\leq\eta r_{0}^{2}$, we see
that on $B(x_{0},2r_{0})\backslash B(x_{0},r_{0})$

$u_{s}\geq\eta r_{0}^{2}-s\geq 0,$

hence we have $\Omega_{s}\subset B(x_{0},r_{0})$. We solve the following real
Monge-Ampère equation on the Euclidean ball $B(x_{0},2r_{0})$

${\rm det}\Big{(}\frac{\partial^{2}\psi_{s,\ell}}{\partial x^{i}\partial
x^{j}}\Big{)}=\frac{\tau_{\ell}(-u_{s})}{A_{s,\ell}}e^{2F}{\rm
det}g,\quad\mbox{in }B(x_{0},2r_{0}),$ (12.14)

$\psi_{s,\ell}=0$ on $\partial B(x_{0},2r_{0})$. Here $\psi_{s,\ell}$ is
strictly convex in the ball $B(x_{0},2r_{0})$ and

$A_{s,\ell}=\int_{B(x_{0},2r_{0})}\tau_{\ell}(-u_{s})e^{2F}({\rm det}g)dx>0$

is chosen so that $\int_{B(x_{0},2r_{0})}{\rm
det}\Big{(}\frac{\partial^{2}\psi_{s,\ell}}{\partial x^{i}\partial
x^{j}}\Big{)}dx=1$. Note that as $\ell\to\infty$

$A_{s,\ell}\to A_{s}:=\int_{\Omega_{s}}(-u_{s})e^{2F}{\rm det}g$

and $A_{s}\leq C(n,g)K\leq\frac{1}{2}C_{1}$, where $C_{1}>0$ depends only on
$n,g$ and $K$. Hence for all $\ell$ sufficiently large which we always assume,
we have

$A_{s,\ell}\leq C_{1}.$ (12.15)

Let $\beta_{n}$ be the volume of the unit ball in ${\bf R}^{2n}$.

###### Lemma 6

There exist a constant $C_{2}=C_{2}(n)>0$ such that

$-\inf_{B(x_{0},2r_{0})}\psi_{s,\ell}\leq C_{2}r_{0},\quad{\rm
sup}_{B(x_{0},r_{0})}|\nabla\psi_{s,\ell}|\leq C_{2}.$

Proof. Since by the definition of $\psi_{s,\ell}$, $\int_{B(x_{0},2r_{0})}{\rm
det}\Big{(}\frac{\partial^{2}\psi_{s,\ell}}{\partial x^{i}\partial
x^{j}}\Big{)}dx=1$, so it follows from the standard ABP maximum principle

$-\inf_{B(x_{0},2r_{0})}\psi_{s,\ell}\leq-\inf_{\partial
B(x_{0},2r_{0})}\psi_{s,\ell}+\frac{4r_{0}}{\beta_{n}}\Big{[}\int_{B(x_{0},2r_{0})}{\rm
det}\Big{(}\frac{\partial^{2}\psi_{s,\ell}}{\partial x^{i}\partial
x^{j}}\Big{)}dx\Big{]}^{1/m}=\frac{4}{\beta_{n}}r_{0}.$

To see the second inequality, for any fixed point $x\in B(x_{0},r_{0})$,
denote $V=\frac{D\psi_{s,\ell}(x)}{|D\psi_{s,\ell}(x)|}$ to be the unit vector
in the direction of $D\psi_{s,\ell}(x)$ (if $D\psi_{s,\ell}(x)=0$, there is
nothing to prove, so here we assume $D\psi_{s,\ell}(x)\neq 0$). Then clearly
$|D\psi_{s,\ell}(x)|=D\psi_{s,\ell}(x)\cdot V$. Consider the half line $L:$
$0\leq t\mapsto x+tV$ which intersects $\partial B(x_{0},r_{0})$ and $\partial
B(x_{0},2r_{0})$ at $L(t_{1})$, $L(t_{2})$, respectively. We have
$t_{2}-t_{1}\geq r_{0}$ and
$0>\psi_{s,\ell}(L(t_{1}))\geq-\frac{4}{\beta_{n}}r_{0}$ and
$\psi_{s,\ell}(L(t_{2}))=0$. Then by the convexity of the function
$t\mapsto\psi_{s,\ell}(L(t))$, we have

$|D\psi_{s,\ell}(x)|=\psi_{s,\ell}(L(t))^{\prime}|_{t=0}\leq\frac{\psi_{s,\ell}(L(t_{2}))-\psi_{s,\ell}(L(t_{1}))}{t_{2}-t_{1}}\leq\frac{4}{\beta_{n}}.$

Taking supremum over all $x\in B(x_{0},r_{0})$ finishes the proof of the
lemma. Q.E.D.

Take positive constants

$\Lambda=\frac{2n}{1+2n}(10C_{J}C_{2})^{2n+1}A_{s,\ell},\quad\varepsilon=\Big{(}\frac{2n+1}{2n}\Big{)}^{\frac{2n}{2n+1}}A_{s,\ell}^{\frac{1}{2n+1}}$
(12.16)

where $C_{2}>0$ is the constant in Lemma 6 and $C_{J}>0$ as in (12.11). We
observe that by (12.15), it holds that $\Lambda$ is bounded above by the
uniform constant $\frac{2n}{1+2n}(10C_{J}C_{2})^{2n+1}C_{1}$.

Define a function $\Phi$ on $B(x_{0},2r_{0})$ by

$\Phi(x)=-\varepsilon(-\psi_{s,\ell}(x)+\Lambda)^{\frac{2n}{2n+1}}-u_{s}(x),\quad\forall
x\in B(x_{0},2r_{0}).$ (12.17)

We claim that $\Phi\leq 0$ on this ball. As a continuous function, $\Phi$
achieves its maximum at some point $x_{\max}\in{\overline{B(x_{0},2r_{0})}}$.
If $x_{\max}\not\in\Omega_{s}$, then by the definition of $\Omega_{s}$,
clearly we have $\Phi(x_{\max})<0$. So we assume $x_{\max}\in\Omega_{s}\subset
B(x_{0},r_{0})$. By the maximum principle, it follows that
$\frac{\partial\Phi}{\partial x^{q}}\Big{|}_{x_{\max}}=0$ and
$\frac{\partial^{2}\Phi}{\partial x^{i}\partial x^{j}}\Big{|}_{x_{\max}}\leq
0$. Hence at $x_{\max}$

$\Delta_{\tilde{g}}\Phi=\tilde{g}^{ij}\frac{\partial^{2}\Phi}{\partial
x^{i}\partial
x^{j}}-\tilde{g}^{ij}\tilde{\Gamma}_{ij}^{q}\frac{\partial\Phi}{\partial
x^{q}}=\tilde{g}^{ij}\frac{\partial^{2}\Phi}{\partial x^{i}\partial x^{j}}\leq
0.$

We then calculate at $x_{\max}$.

$\displaystyle 0$ $\displaystyle\geq$
$\displaystyle\Delta_{\tilde{g}}\Phi=\frac{2n\varepsilon}{2n+1}(-\psi_{s,\ell}+\Lambda)^{-\frac{1}{2n+1}}\Delta_{\tilde{g}}\psi_{s,\ell}-\Delta_{\tilde{g}}\varphi-\eta\Delta_{\tilde{g}}|x|^{2}$
(12.18)
$\displaystyle+\frac{2n\varepsilon}{(2n+1)^{2}}(-\psi_{s,\ell}+\Lambda)^{-\frac{2n+2}{2n+1}}|\nabla\psi_{s,\ell}|^{2}_{\tilde{g}}$
$\displaystyle\geq$
$\displaystyle\frac{2n\varepsilon}{2n+1}(-\psi_{s,\ell}+\Lambda)^{-\frac{1}{2n+1}}\Delta_{\tilde{g}}\psi_{s,\ell}-2n+{\rm
tr}_{\tilde{g}}g-\eta\Delta_{\tilde{g}}|x|^{2}.$

We first look at the term $-\eta\Delta_{\tilde{g}}|x|^{2}$ in (12.18). It
satisfies

$-\eta\Delta_{\tilde{g}}|x|^{2}=-2\eta\tilde{g}^{ij}\delta_{ij}+2\eta\tilde{g}^{ij}\tilde{\Gamma}_{ij}^{q}x^{q}\geq-4\eta{\rm
tr}_{\tilde{g}}g-2\eta C_{J}r_{0}{\rm tr}_{\tilde{g}}g\geq-\frac{1}{10}{\rm
tr}_{\tilde{g}}g,$ (12.19)

where $C_{J}>0$ is the uniform constant in (12.11) and we have chosen $\eta>0$
such that

$\eta(4+2C_{J}r_{0})=1/10$

and this fixes the uniform constant $\eta$. We will denote
$D^{2}_{ij}\psi_{s,\ell}=\frac{\partial^{2}\psi_{s,\ell}}{\partial
x^{i}\partial x^{j}}$ to be (Euclidean) Hessian of the function
$\psi_{s,\ell}$. For the first term in (12.18), we have

$\displaystyle\frac{2n\varepsilon}{2n+1}(-\psi_{s,\ell}+\Lambda)^{-\frac{1}{2n+1}}\Delta_{\tilde{g}}\psi_{s,\ell}$
(12.20) $\displaystyle=$
$\displaystyle\frac{2n\varepsilon}{2n+1}(-\psi_{s,\ell}+\Lambda)^{-\frac{1}{2n+1}}\tilde{g}^{ij}\Big{(}\frac{\partial^{2}\psi_{s,\ell}}{\partial
x^{i}\partial
x^{j}}\Big{)}+\frac{2n\varepsilon}{2n+1}(-\psi_{s,\ell}+\Lambda)^{-\frac{1}{2n+1}}\tilde{g}^{ij}\tilde{\Gamma}_{ij}^{q}\frac{\partial\psi_{s,\ell}}{\partial
x^{q}}$ $\displaystyle\geq$
$\displaystyle\frac{4n^{2}\varepsilon}{2n+1}(-\psi_{s,\ell}+\Lambda)^{-\frac{1}{2n+1}}\Big{(}\frac{{\rm
det}D^{2}\psi_{s,\ell}}{{\rm
det}\tilde{g}}\Big{)}^{1/2n}-\frac{2n\varepsilon\Lambda^{-\frac{1}{2n+1}}}{2n+1}C_{J}|\frac{\partial\psi_{s,\ell}}{\partial
x^{q}}|\cdot{\rm tr}_{\tilde{g}}g$ $\displaystyle\geq$
$\displaystyle\frac{4n^{2}\varepsilon}{2n+1}(-\psi_{s,\ell}+\Lambda)^{-\frac{1}{2n+1}}\Big{(}\frac{-u_{s}}{A_{s,\ell}}\Big{)}^{1/2n}-\frac{2n\varepsilon\Lambda^{-\frac{1}{2n+1}}}{2n+1}C_{J}C_{2}\cdot{\rm
tr}_{\tilde{g}}g$ $\displaystyle\geq$
$\displaystyle\frac{4n^{2}\varepsilon^{1+\frac{1}{2n}}}{(2n+1)A_{s,\ell}^{1/2n}}\Big{(}\frac{-u_{s}}{\varepsilon(-\psi_{s,\ell}+\Lambda)^{2n/(2n+1)}}\Big{)}^{1/2n}-\frac{1}{10}{\rm
tr}_{\tilde{g}}g$ $\displaystyle=$ $\displaystyle
2n\Big{(}\frac{-u_{s}}{\varepsilon(-\psi_{s,\ell}+\Lambda)^{2n/(2n+1)}}\Big{)}^{1/2n}-\frac{1}{10}{\rm
tr}_{\tilde{g}}g.$

Combining the equations (12.20), (12.19), (12.18), we see that at $x_{\max}$

$0\geq
2n\Big{(}\frac{-u_{s}}{\varepsilon(-\psi_{s,\ell}+\Lambda)^{2n/(2n+1)}}\Big{)}^{1/2n}-2n+\frac{4}{5}{\rm
tr}_{\tilde{g}}g$

from which we easily derive that
$\frac{(-u_{s})}{\varepsilon(-\psi_{s,\ell}+\Lambda)^{2n/(2n+1)}}<1$, that is,
$\Phi|_{x_{\max}}<0$. Hence we finish the proof of the claim that $\Phi\leq
0$. In particular on $\Omega_{s}$ it holds that

$-u_{s}\leq
C(n)A_{s,\ell}^{\frac{1}{2n+1}}(-\psi_{s,\ell}+\Lambda)^{\frac{2n}{2n+1}}\leq
C_{3}A_{s,\ell}^{\frac{1}{2n+1}},$ (12.21)

for some $C_{3}>0$ that depends on $n,g,J,K$. Here we have applied Lemma 6 to
see that $|\psi_{s,\ell}|\leq C_{2}r_{0}$ and the fact that $\Lambda$ is a
uniformly bounded constant. Letting $\ell\to\infty$ we conclude from (12.21)
that

$-u_{s}\leq C_{3}A_{s}^{\frac{1}{2n+1}},$ (12.22)

Integrating both sides of (12.22) agaist the measure $e^{2F}({\rm det}g)dx$
over $\Omega_{s}$, we get

$A_{s}=\int_{\Omega_{s}}(-u_{s})e^{2F}({\rm det}g)dx\leq
C_{3}A_{s}^{\frac{1}{2n+1}}\int_{\Omega_{s}}e^{2F}({\rm det}g)dx.$ (12.23)

So we have

$A_{s}\leq C_{3}^{\frac{2n+1}{2n}}\Big{(}\int_{\Omega_{s}}e^{2F}({\rm
det}g)dx\Big{)}^{1+\frac{1}{2n}}=C_{4}\phi(s)^{1+\frac{1}{2n}},$ (12.24)

where we denote $\phi(s)=\Big{(}\int_{\Omega_{s}}e^{2F}({\rm
det}g)dx\Big{)}^{1+\frac{1}{2n}}$. On the other hand, for any $0<t<s$, on the
open set $\Omega_{s-t}$ we have

$u_{s}(x)=u_{s-t}(x)-t<-t,\mbox{ i.e. }-u_{s}(x)>t$

It is elementary that

$A_{s}\geq\int_{\Omega_{s-t}}(-u_{s})e^{2F}({\rm det}g)dx\geq t\phi(s-t).$

Combining the above, we see that

$t\phi(s-t)\leq C_{4}\phi(s)^{1+\frac{1}{2n}},\quad\forall\,0<t<s\leq s_{0}.$
(12.25)

It is not hard to see that $\phi(s)$ is an increasing and continuous function
in $s\in(0,s_{0}]$ and $\phi(s)>0$ for any $s\in(0,s_{0}]$ and $\lim_{s\to
0^{+}}\phi(s)=0$. We can apply a version of De Giorgi’s lemma to show that
$\frac{2C_{4}}{1-2^{-1/2n}}\phi(s_{0})^{1/2n}\geq s_{0}$. Hence there is a
uniform constant $c_{0}>0$ such that

$\phi(s_{0})\geq c_{0}>0.$ (12.26)

Applying (12.24) with $s=s_{0}$, we obtain $A_{s_{0}}\leq C_{5}$ for a
constant $0<C_{5}=C_{4}(2^{2n}\beta_{n}K)^{1+\frac{1}{2n}}$. From the
definition of $A_{s_{0}}$, we derive that

$(-\varphi(x_{0}))\cdot\phi(s_{0})\leq
s_{0}\phi(s_{0})+\int_{\Omega_{s_{0}}}(-\varphi)e^{2F}({\rm det}g)dx+C_{5}.$
(12.27)

The equation (12.5) then implies that

$-{\rm inf}_{X}\varphi=-\varphi(x_{0})\leq
C_{6}+C_{7}\int_{X}(-\varphi)e^{2F}dV_{g},$ (12.28)

for some constant $C_{7}>0$. So we have proved the inequality for the solution
$\varphi$ to the equation (12.2)

${\rm sup}_{X}|\varphi|\leq C_{8}(1+\|\varphi\|_{L^{1}(X,e^{2F}dV_{g})}).$
(12.29)

This finishes the proof of Theorem 16. Q.E.D.

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Department of Mathematics & Computer Science, Rutgers University, Newark, NJ
07102 USA

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Department of Mathematics, Columbia University, New York, NY 10027 USA

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