4 Monge-Ampère equations [057P]

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4 Monge-Ampère equations

In this and the next section, we apply Theorems 1 and 2 to the specific cases of the Monge-Ampère and Hessian equations on a compact Kähler manifold $(X,\omega_{X})$ .

We begin by noting that the structural condition (1.4) holds for many equations and is usually easy to check:

Lemma 4

Assume that $f:{\bf R}^{n}\to{\bf R}_{+}$ is a concave and homogeneous function of degree one, which satisfies $\frac{\partial f(\lambda)}{\partial\lambda_{j}}>0$ for any $\lambda$ in an admissible cone $\Gamma\subset{\bf R}^{n}$ . Assume that there is a $\gamma>0$ such that

f(\mu)\geq n\gamma^{1/n}(\prod_{j}\mu_{j})^{1/n},\quad\mbox{for all }\mu\in\Gamma_{n}:=\{\lambda\in{\bf R}^{n}:\lambda_{1}>0,\ldots,\lambda_{n}>0\}.

(4.1)

Then $f$ satisfies the structural condition (1.4).

Proof. By the concavity of $f$ on $\Gamma$ , for any $\lambda,\mu\in\Gamma$ we have

f(\mu)\leq f(\lambda)+\sum_{j=1}^{n}(-\lambda_{j}+\mu_{j})\frac{\partial f(\lambda)}{\partial\lambda_{j}}=\sum_{j=1}^{n}\mu_{j}\frac{\partial f(\lambda)}{\partial\lambda_{j}},

(4.2)

where we have used the homogeneity of degree one assumption on $f$ , which implies that $\sum_{j}\lambda_{j}\frac{\partial f(\lambda)}{\partial\lambda_{j}}=f(\lambda)$ . Taking the infimum of the right hand side of (4.2) over all $\mu\in\Gamma_{n}$ with $\prod_{j=1}^{n}\mu_{j}=1$ , by the arithmetic-geometric inequality and the assumption (4.1) on $f$ , we get

\prod_{j=1}^{n}\frac{\partial f(\lambda)}{\partial\lambda_{j}}\geq n^{-n}\big\{\inf_{\mu\in\Gamma_{n},\prod_{j}\mu_{j}=1}f(\mu)\big\}^{n}\geq\gamma>0.

The desired inequality (1.4) follows from this inequality upon diagonalizing the matrix $h$ . The proof of Lemma 4 is complete.

It follows immediately from this lemma that the functions $f(\lambda)=(\prod_{j=1}^{n}\lambda_{j})^{1\over n}$ , $f(\lambda)=\sigma_{k}(\lambda)^{1\over k}$ , and $f(\lambda)=({\sigma_{k}(\lambda)\over\sigma_{\ell}(\lambda)})^{1\over k-\ell}+c\sigma_{p}(\lambda)^{\frac{1}{p}}$ for $c>0$ , $n\geq k\geq\ell\geq 1$ , $n\geq p\geq 1$ , corresponding respectively to the Monge-Ampère, the Hessian, and the quotient Hessian equations, all satisfy the structural condition (1.4), and the constant $\gamma$ depends only on the given numbers $n,k,\ell,p,c>0$ , and the admissible cone $\Gamma=\Gamma_{k}=\{\lambda\in{\bf R}^{n}:\sigma_{1}(\lambda)>0,\ldots,\sigma_{k}(\lambda)>0\}$ . The last equation appeared recently in [8].

In this section, we focus on the Monge-Ampère equation. To discuss the underlying geometry, it is convenient to rewrite it in the more usual form

\displaystyle(\omega_{t}+i\partial\bar{\partial}\varphi_{t})^{n}=c_{t}^{n}e^{nF_{t}}\omega_{X}^{n},\quad\lambda[h_{t,\varphi_{t}}]\in\Gamma,\quad\sup_{X}\varphi_{t}=0,

(4.3)

where $\omega_{t}=t\omega_{X}+\chi$ is the family of degenerating background metrics. To apply Theorem 2, we need to control the ratio $c_{t}^{n}/V_{t}$ and the energy $E_{t}$ . This can be readily done using the following two easy lemmas:

Lemma 5

Let $V=\int_{X}\omega_{X}^{n}$ be the volume of $(X,\omega_{X})$ , then

V^{-1}=\frac{c_{t}^{n}}{V_{t}},\quad\forall t\in(0,1].

Proof. Integrating both sides of (4.3), we get

c_{t}^{n}V=c_{t}^{n}\int_{X}e^{nF_{t}}\omega_{X}^{n}=\int_{X}(\omega_{t}+i\partial\bar{\partial}\varphi_{t})^{n}=\int_{X}\omega_{t}^{n}=V_{t}.

Lemma 6

There is a uniform constant $C>0$ depending only on $n,\|e^{nF_{t}}\|_{L^{1}(\,{\rm log}\,L)^{1}(\omega_{X}^{n})}$ , $\omega_{X}$ , $\chi$ such that for all $t\in(0,1]$

\displaystyle E_{t}(\varphi_{t})\leq C.

(4.4)

Proof. Recall that

E_{t}(\varphi_{t})=\frac{c_{t}^{n}}{V_{t}}\int_{X}(-\varphi_{t})e^{nF_{t}}\omega_{X}^{n}=\frac{1}{V_{t}}\int_{X}(-\varphi_{t})\omega_{\varphi_{t}}^{n},

so that it suffices to show that $\frac{1}{V_{t}}\int_{X}(-\varphi_{t})\omega_{\varphi_{t}}^{n}\leq C$ , $\forall t\in(0,1].$ We may assume without loss of generality that $\chi\leq\omega_{X}$ , so the $\omega_{t}$ -plurisubharmonic function $\varphi_{t}$ is also $2\omega_{X}$ -plurisubharmonic and by the $\alpha$ -invariant estimate, there is an $\alpha_{0}=\alpha_{0}(X,\omega_{X})>0$ such that

\frac{1}{V_{t}}\int_{X}\,{\rm exp}\,\Big(-\,{\rm log}\,\frac{\omega_{\varphi_{t}}^{n}}{\omega_{X}^{n}}-\alpha_{0}\varphi_{t}\Big)\omega_{\varphi_{t}}^{n}=\frac{1}{V_{t}}\int_{X}e^{-\alpha_{0}\varphi_{t}}\omega_{X}^{n}\leq\frac{C(n,\omega_{X})}{V_{t}}

By Jensen’s inequality it follows that

\frac{1}{V_{t}}\int_{X}\Big(-\,{\rm log}\,\frac{\omega_{\varphi_{t}}^{n}}{\omega_{X}^{n}}-\alpha_{0}\varphi_{t}\Big)\omega_{\varphi_{t}}^{n}\leq\,{\rm log}\,C-\,{\rm log}\,V_{t}

which implies that

$\displaystyle\frac{1}{V_{t}}\int_{X}(-\alpha_{0}\varphi_{t})\omega_{\varphi_{t}}^{n}$	$\displaystyle\leq$	$\displaystyle\frac{1}{V_{t}}\int_{X}\,{\rm log}\,(e^{nF_{t}}c_{t}^{n})\omega_{\varphi_{t}}^{n}+\,{\rm log}\,C-\,{\rm log}\,V_{t}$
	$\displaystyle=$	$\displaystyle\int_{X}(nF_{t})e^{nF_{t}}\frac{c_{t}^{n}}{V_{t}}\omega_{X}^{n}+\,{\rm log}\,C+\,{\rm log}\,\frac{c^{n}_{t}}{V_{t}}$
	$\displaystyle\leq$	$\displaystyle C\\|e^{nF_{t}}\\|_{L^{1}(\,{\rm log}\,L)^{1}(\omega_{X})}+C,$

from which the estimate follows since $c^{n}_{t}$ and $V_{t}$ are uniformly equivalent by Lemma 5.

Theorem 2 together with the preceding lemmas implies at once the following basic estimates of Kolodziej [14], Eyssidieux, Guedj, and Zeriahi [12], and Demailly and Pali [9]:

Theorem 4

Consider the above family (4.3) of complex Monge-Ampère equations, with respect to the degenerating background metrics $\omega_{t}$ , $t\in(0,1]$ . Fix any $q>n$ . If $\varphi_{t}$ is a family of $C^{2}$ solution, normalized by $\sup_{X}\varphi_{t}=0$ , and if $\|e^{nF_{t}}\|_{L^{1}(\,{\rm log}\,L)^{q}(\omega_{X}^{n})}$ is uniformly bounded in $t$ , then $\|\varphi_{t}\|_{L^{\infty}(X)}$ is uniformly bounded in $t$ as well.

In many applications, the $(1,1)$ -form $\chi$ is chosen to be $\chi=\pi^{*}\omega_{Y}$ , where $\pi:X\to Y$ is a holomorphic map between Kähler manifolds and $\omega_{Y}$ is a Kähler metric on $Y$ .