A Priori Estimates of the Degenerate Monge-Ampere Equation on Kahler Manifolds of Nonnegative Bisectional Curvature

Abstract

The regularity theory of the degenerate complex Monge-Ampère equation is studied. The equation is considered on a closed compact Kähler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a solution $\varphi$ of the degenerate complex Monge-Ampère equation $\det (g_{i\bar{j}}+{\varphi }_{i\bar{j}})=f\det (g_{i\bar{j}})$, it is shown that the Laplacian of $\varphi$ can be controlled by a constant depending on $(M,g)$, $\sup f$, and ${\inf }_M\Delta f^{1/(m-1)}$.

arXiv commentTo appear in Mathematical Research Letters