A free boundary Monge-Ampère equation and applications to complete Calabi-Yau metrics

Abstract

Let $P$ be a convex body containing the origin in its interior. We study a real Monge-Ampère equation with singularities along $\del P$ which is Legendre dual to a certain free boundary Monge-Ampère 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 $\del P$.