An Optimal Transport Approach to Monge–Ampère Equations on Compact Hessian Manifolds
Artikel i vetenskaplig tidskrift, 2018

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.

Monge-Ampère equations

Affine geometry

Hessian manifolds

Optimal transport

Författare

Jakob Hultgren

Chalmers, Matematiska vetenskaper, Algebra och geometri

Magnus Önnheim

Chalmers, Matematiska vetenskaper, Algebra och geometri

Journal of Geometric Analysis

1050-6926 (ISSN)

Fundament

Grundläggande vetenskaper

Ämneskategorier

Geometri

Matematisk analys

DOI

10.1007/s12220-018-0068-5

Mer information

Skapat

2018-08-31