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arXiv · 2402.10111

A free boundary Monge-Amp\`ere equation and applications to complete Calabi-Yau metrics

Tristan C. Collins·Freid Tong·Shing-Tung Yau
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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 $\del P$.

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Tristan C. Collins, Freid Tong, Shing-Tung Yau. 2024-02-15. A free boundary Monge-Amp\`ere equation and applications to complete Calabi-Yau metrics. https://arxiv.org/abs/2402.10111

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