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A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics (2402.10111v1)
Published 15 Feb 2024 in math.DG
Abstract: Let $P$ be a convex body containing the origin in its interior. We study a real Monge-Amp`ere equation with singularities along $\del P$ which is Legendre dual to a certain free boundary Monge-Amp`ere equation. This is motivated by the existence problem for complete Calabi-Yau metrics on log Calabi-Yau pairs $(X, D)$ with $D$ an ample, simple normal crossings divisor. We prove the existence of solutions in $C{\infty}(P)\cap C{1,\alpha}(\overline{P})$, and establish the strict convexity of the free boundary. When $P$ is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension $1$ faces of $\del P$.