The main goal of this paper is to answer a long-standing question in the theory of complex Monge-Ampère equations and its applications to complex geometry, namely whether sharp $L^{\infty}$ estimates can be established by PDE methods and without pluripotential theory. We shall see that the answer is affirmative and, as may have been anticipated, the PDE proof also gives new estimates as well as extensions to many other fully non-linear equations.

$L^{\infty}$ estimates have a uniquely storied history in the theory of complex Monge-Ampère equations. Early on, they were recognized as the defining difficulty in the problem of finding Kähler-Einstein metrics. Yau’s introduction in 1978 of Moser iteration was the key step in his solution of the Calabi conjecture, and it ushered in a new era for complex Monge-Ampère equations [25]. Yau’s Moser iteration method works for equations whose right hand sides are in $L^{q}$ for $q>n$, where $n$ is the complex dimension of the underlying space. The next major advance was the 1998 result of Kolodziej [14], which established $L^{\infty}$ estimates for the solution when the right hand side is in $L^{q}$, for $q>1$. This improvement in the range of $q$ is no mere technicality, and it has a profound geometric significance: that $L^{\infty}$ estimates fail for $q=1$ is indicative of the necessity of stability conditions in the Kähler-Ricci flow [16], while the cases $1<q\leq n$ are needed in a wide range of problems, including singular Kähler-Einstein metrics on manifolds of general type [12, 9], the analytic minimal model program [18], and degenerating Calabi-Yau metrics [21]. Kolodziej’s method of proof relied heavily on the pluripotential theory developed in the late 1970’s by Bedford and Taylor [3, 4]. Actually, for many applications to geometry, an extension of Kolodziej’s results to the more general case of degenerating background metrics is necessary. Such an extension was developed in 2007 independently by Demailly and Pali [9] and Eyssidieux, Guedj, and Zeriahi [12], and pluripotential theory continued to be essential.

An immediate question for the theory of geometric partial differential equations is whether the above $L^{\infty}$ estimates can be derived by PDE methods, instead of pluripotential theory which is specific to Monge-Ampère equations. This question gained considerable attention over the years, as more and more fully non-linear equations without corresponding pluripotential theory emerged in complex differential geometry. A completely different proof for right hand sides in $L^{q}$ with $q>2$ was found in 2011 by Blocki [5], using the Alexandrov-Bakelman-Pucci (ABP) maximum principle.¹¹ 1 The possibility of applying ABP maximum principle to the complex Monge-Ampère equation were suggested a while ago by S.Y. Cheng and S.T. Yau. His methods turn out to be remarkably powerful, and have since been applied successfully to many problems, including subsolutions [19, 17], equations with gradient terms [23], and the constant scalar curvature problem [7]. Even so, extensions to degenerating backgrounds as well as the full range $q>1$ remained out of reach. In a different direction, a PDE proof of $L^{\infty}$ estimates for the complex Monge-Ampère equation was obtained by J.X. Wang, X.J. Wang, and B. Zhou [24] for domains in ${\bf C}^{n}$. However, their methods do not appear adaptable to the compact manifold case, even in the simplest case when the background Kähler metric is fixed. Thus a fully effective approach to $L^{\infty}$ estimates remained an open question.

The PDE approach to $L^{\infty}$ estimates which we present in this paper combines the methods of Wang, Wang, and Zhou [24] with a fundamental new idea of Chen and Cheng [7] in their recent work on constant scalar curvature Kähler metrics, namely to compare the given equation with an auxiliary complex Monge-Ampère equation. A key novelty in our paper resides in the choice of auxiliary complex Monge-Ampère equation, as well as of the test function $\Phi$ for the comparison. We now formulate our main results.

Let $X$ be a compact Kähler manifold without boundary of dimension $n$, and $\omega_{X}$ its Kähler form. If $\varphi$ is a real smooth function on $X$, we let $\omega_{\varphi}=\omega_{X}+i\partial\bar{\partial}\varphi$, and let $h_{\varphi}$ be the corresponding endomorphism relative to the metric $\omega_{X}$. Explicitly, if we write $\omega_{X}=ig_{\bar{k}j}dz^{j}\wedge d\bar{z}^{k}$ in local holomorphic coordinates, then $(h_{\varphi})^{j}{}_{k}=g^{j\bar{m}}(\omega_{\varphi})_{\bar{m}k}$. Let $\lambda[h_{\varphi}]$ be the vector of eigenvalues of $h_{\varphi}$, and consider the non-linear partial differential equation

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

(1.1)

for a given function $f(\lambda)$ and real function $F$ normalized such that $\int_{X}e^{nF}\omega_{X}^{n}=V=\int_{X}\omega_{X}^{n}$. Here the function $f(\lambda)$ is assumed to be invariant under permutations of the components of $\lambda$, and defined on an open cone $\Gamma\subset\{\lambda:\lambda_{1}+\ldots+\lambda_{n}>0\}$ with vertex at the origin and containing the first octant $\{\lambda:\lambda_{1}>0,\ldots,\lambda_{n}>0\}$. We assume throughout $f$ is elliptic in the sense that $\frac{\partial f(\lambda)}{\partial\lambda_{j}}>0$ for any $\lambda\in\Gamma$, and that

$\displaystyle f(r\lambda)=r\,f(\lambda),\quad r>0,\ \lambda\in\Gamma,$

(1.2)

which can be viewed as a normalization, as the same homogeneous equation can be expressed with many different functions $f(\lambda)$. We shall need the $L^{1}(\,{\rm log}\,L)^{p}$ norm of $e^{nF}$ with respect to the measure $\omega_{X}^{n}$, which can be recognized as a generalized entropy ${\rm Ent}_{p}(F)$,

$\displaystyle{\rm Ent}_{p}(F)={1\over V}\int_{X}e^{nF}|F|^{p}\omega_{X}^{n}={1\over n^{p}V}\|e^{nF}\|_{L^{1}(\,{\rm log}\,L)^{p}}.$

(1.3)

We observe that many equations satisfy the structural condition (1.4). They include the Monge-Ampère equation $f(\lambda)=(\prod_{j=1}^{n}\lambda_{j})^{1\over n}$, more generally Hessian equations $f(\lambda)=\sigma_{k}(\lambda)^{1/k}$, $k=1,\ldots,n$, where $\sigma_{k}(\lambda)$ is the $k$-th symmetric function, and quotient Hessian equations such as $f(\lambda)=\big({\sigma_{k}\over\sigma_{l}}\big)^{1/(k-l)}+c\sigma_{m}^{1/m}$ for some $c>0$. Applying Theorem 1 to the Monge-Ampère equation, we obtain immediately Kolodziej’s [14] sharp $L^{\infty}$ bounds. Applying it to Hessian equations, we obtain the $L^{\infty}$ bounds of Dinew and Kolodziej [11]. In fact, our result is stronger, as the bound in [11] requires that $\,{\rm exp}\,(nF)$ be in $L^{q}$ for $q>1$, while we only need that $\,{\rm exp}\,(nF)$ be in $L^{1}(\,{\rm log}\,L)^{p}$ for $p>n$. Beyond these cases, the $L^{\infty}$ estimates in Theorem 1 are new, and appear to be the first obtained in any generality for fully non-linear equations from Kähler geometry.

We discuss now estimates, particularly important for many geometric applications, where the background metric is allowed to degenerate. It is convenient to set up the equation as follows. Let $(X,\omega_{X})$ be a compact Kähler manifold of dimension $n$ as before, and let $\chi$ be a fixed closed and non-negative $(1,1)$-form. For $t\in(0,1]$, set

$\displaystyle\omega_{t}=\chi+t\omega_{X}$

(1.6)

and for each $\varphi\in C^{2}(X)$, let $\omega_{t,\varphi}=\omega_{t}+i\partial\bar{\partial}\varphi$, $h_{t,\varphi}$ be the corresponding endomorphism relative to the metric $\omega_{X}$, and $\lambda[h_{t,\varphi}]$ be the vector of eigenvalues of $h_{t,\varphi}$. Consider the family of non-linear partial differential equations

$\displaystyle f(\lambda[h_{t,\varphi_{t}}])=c_{t}\,e^{F_{t}},\quad{\rm sup}_{X}\varphi_{t}=0,\quad\lambda[h_{t,\varphi_{t}}]\in\Gamma,\quad t\in(0,1]$

(1.7)

for given function $f(\lambda)$, real functions $F_{t}$, and positive constant coefficients $c_{t}$. Here the functions $F_{t}$ are normalized by $\int_{X}e^{nF_{t}}\omega_{X}^{n}=\int_{X}\omega_{X}^{n}$, so that the constants $c_{t}$ are determined. Denote by $V_{t}$ the volume of the metric $\omega_{t}$, $V_{t}=\int_{X}\omega_{t}^{n}$, and define the energy $E_{t}(\varphi)$ by

$\displaystyle E_{t}(\varphi)=\frac{1}{V_{t}}\int_{X}(-\varphi)f(\lambda([h_{t,\varphi}])^{n}\omega_{X}^{n}.$

(1.8)

For functions $\varphi_{t}$ solving the equation (1.7), we obviously have

$\displaystyle E_{t}(\varphi_{t})=\frac{c^{n}_{t}}{V_{t}}\int_{X}(-\varphi_{t}){\rm exp}(nF_{t})\omega_{X}^{n}.$

(1.9)

All three quantities in (1.11) have attractive interpretations. We have already noted from (1.3) that ${\rm Ent}_{p}(F_{t})$ is a generalized entropy. The quantity $c_{t}^{n}\over V_{t}$ can be viewed as a relative volume, and a substitute for a cohomological constraint when dealing with general equations $f(\lambda)$. The quantity $E_{t}(\varphi_{t})$ is clearly an energy functional, as it reduces to the Dirichlet integral in the case of the Laplacian on surfaces.

As a special case, Theorem 2 applied to the Monge-Ampère equation gives back immediately the $L^{\infty}$ estimates of Eyssidieux, Guedj, Zeriahi [12], and Demailly, Pali [9], in the full generality of degenerating background metrics. The point here is that the relative volumes $c_{t}^{n}/V_{t}$ must be $1/V$ because of a cohomological constraint, and it follows easily from Jensen’s inequality that the energies $E_{t}$ are uniformly bounded (see the fuller discussion in Theorem 4 in §4 below). Thus Theorem 2 gives the pure PDE proof of these $L^{\infty}$ estimates that we sought. In fact, it is particularly simple as the ABP maximum principle is not even needed.

More generally, as long as the admissible cone $\Gamma$ is the one corresponding to PSH functions, we can obtain uniform bounds for the energies $E_{t}$. For example, this applies to the equations $f(\lambda)=\big(\frac{\sigma_{k}}{\sigma_{l}}\big)^{1/(k-l)}+c\;\sigma_{n}^{1/n}$.

For Hessian equations $f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$, $1\leq k<n$, nothing was known in the case of degenerating background metrics, and Theorem 2 is now the only available result. As we shall see in Theorem 5 in §5 below, if $\|e^{nF_{t}}\|_{L^{p}}$ is uniformly bounded for $p>{n\over k}$, we can bound the energy $E_{t}$ by a multiple of ${c_{t}^{n}V_{t}^{-1}}$. In particular, for big cohomology classes $[\omega_{t}]$, the volumes $V_{t}$ do not tend to $0$, the coefficients $c_{t}^{n}$ are bounded by the entropy, and we obtain then uniform $L^{\infty}$ bounds. Such bounds are completely new for fully non-linear equations, and in particular Theorem 6 is new for Hessian equations.