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An Optimal Transport Approach to Monge-Ampère Equations on Compact Hessian Manifolds

Journal article
Authors Jakob Hultgren
Magnus Önnheim
Published in Journal of Geometric Analysis
Volume 29
Issue 3
Pages 1953–1990
ISSN 1050-6926
Publication year 2019
Published at Department of Mathematical Sciences
Pages 1953–1990
Language en
Links https://doi.org/10.1007/s12220-018-...
Keywords Affine geometry, Hessian manifolds, Monge–Ampère equations, Optimal transport
Subject categories Geometry, Mathematical Analysis

Abstract

In this paper we consider Monge–Ampère equations on compact Hessian manifolds, or equivalently Monge–Ampère equations on certain unbounded convex domains in Euclidean space, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume preserving, i.e., when the manifold is special, the solvability of the corresponding Monge–Ampère equation was first established by Cheng and Yau using the continuity method. In the general case we set up a variational framework involving certain dual manifolds and a generalization of the classical Legendre transform. We give existence and uniqueness results and elaborate on connections to optimal transport and quasi-periodic tilings of convex domains.

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