An optimal transport approach to Monge-Amp\`ere equations on compact Hessian manifolds

TL;DR

This paper develops a variational approach to solve Monge-Ampère equations on compact Hessian manifolds, extending previous results to non-volume-preserving cases and connecting to optimal transport and tilings.

Contribution

It introduces a new variational framework and generalizes the Legendre transform to address Monge-Ampère equations on Hessian manifolds with affine group actions.

Findings

Established existence and uniqueness of solutions in the general case.

Connected Monge-Ampère equations to optimal transport theory.

Linked solutions to quasi-periodic tilings of convex domains.

Abstract

In this paper we consider Monge-Amp\`ere equations on compact Hessian manifolds, or equivalently Monge-Amp\`ere equations on certain unbounded convex domains $\Omega\subseteq \mathbb{R}^n$, 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-Amp\`ere equation was established using the continuity method by Cheng and Yau. 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, elaborate on connections to optimal transport and quasi-periodic tilings of convex domains.