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

Tristan C. Collins, Freid Tong, Shing-Tung Yau

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 $C^{\infty}(P)\cap C^{1,α}(\overline{P})$, 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 $\del P$.

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

Abstract

Let be a convex body containing the origin in its interior. We study a real Monge-Ampère equation with singularities along 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 with an ample, simple normal crossings divisor. We prove the existence of solutions in , and establish the strict convexity of the free boundary. When is a polytope, we obtain an asymptotic expansion for the solution near the interior of the codimension faces of .
Paper Structure (7 sections, 18 theorems, 128 equations)

This paper contains 7 sections, 18 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Geometric motivation
  3. Monge-Ampère functionals and their variations
  4. Proof of Theorem \ref{['thm: main-theorem']}
  5. Solution of a variational problem
  6. Solution of \ref{['eqn: main-equation']}
  7. Regularity

Key Result

Theorem 1.1

Let $P$ be an open convex set that contains the origin, then the equation eqn: main-equation admits a solution $v\in C^{\infty}(P)\cap C^{1, \alpha}(\overline P)$ for some $\alpha>0$.

Theorems & Definitions (38)

  • Remark 1
  • Theorem 1.1
  • Remark 2
  • Corollary 1.1
  • Theorem 1.2
  • Conjecture 1: Liouville Theorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • ...and 28 more