Priori estimates of the degenerate Monge-Ampère equation on Kähler manifolds of non-negative bisectional curvature

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 non-negative orthogonal bisectional curvature of dimension $m$. Given a solution $\phi$ of the degenerate complex Monge-Ampère 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 $\inf_M \Delta f^{1/(m-1)}$.

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Published 13 June 2014