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-Amp\`{e}re equation is studied. The equation is considered on a closed compact K\"{a}hler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a solution $\phi$ of the degenerate complex Monge-Amp\`{e}re equation $\det(g_{i \bar{j}} + \phi_{i \bar{j}}) = f \det(g_{i \bar{j}})$, it is shown that the Laplacian of $\phi$ can be controlled by a constant depending on $(M,g)$, $\sup f$, and $\inf_M \Delta f^{1/(m-1)}$.