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

Abstract

In this paper we consider Monge-Ampère equations on compact Hessian manifolds, or equivalently Monge-Ampère equations on certain unbounded convex domains Ω ⊆ 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-Ampère equation was established using the continuity method in [6]. 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.

On geometry of steady toric Kähler-Ricci solitons

“…The Hessian formula follows immediately from the expression for the Christoffel symbols of Lemma 4.2. The Laplacian formula (18) is then obtained by taking the trace of (17) and using the first differential identity of Lemma 4.1. Now we are ready to prove the key geometric property of the solutions to the Monge-Ampère equation (16).…”

Convergence of the Hesse–Koszul flow on compact Hessian manifolds

“…They used the continuity method for D 0 and the viscosity method for > 0. We also refer to [20] for a recent variational approach with optimal transport point of view.…”