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 \(\partial P\) 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 \(\partial P\).