On the Local Regularity of Homogeneous Complex Monge-Ampère Equations

Abstract

We establish a local higher regularity theorem for the homogeneous complex Monge-Ampere equation on $X\times D$, where $X$ is a complete, possibly noncompact Kahler manifold and $D\subset\mathbb C$ is the unit disc. For a bounded weak solution given by the Perron envelope, we prove higher regularity on a smaller spatial domain when the smooth boundary data are locally small. The geometric hypotheses require uniformly convex local Kahler potentials and a local barrier threshold sufficiently large relative to the global $L^\infty$ bound of the solution. We also obtain quantitative local Holder estimates. The proof combines a perturbative construction of local holomorphic disc foliations with a construction of global subsolutions using convexity and barriers. A key analytic difficulty is the loss of Holder regularity in the construction of holomorphic disc families, which we address using BMO estimates and Nash-Moser-Zehnder theorems. Comparison along the discs identifies the local smooth solution by holomorphic discs foliation with the global weak solution. We apply the theorem and its proof to compact Kahler manifolds and to ALE, ALF, and asymptotically complex hyperbolic (ACH) ends. Under the stated geometric and decay assumptions, we obtain higher regularity of each prescribed order sufficiently far out, together with weighted Holder estimates preserving the boundary decay.