Monge-Amp\`ere equations on compact Hessian manifolds

TL;DR

This paper develops a comprehensive framework for solving degenerate Monge-Ampère equations on compact Hessian manifolds, extending classical results and providing tools potentially applicable to singular Hessian varieties and the Strominger-Yau-Zaslow conjecture.

Contribution

It introduces new compactness and comparison principles for Monge-Ampère equations on Hessian manifolds and generalizes existing solutions to include arbitrary probability measures.

Findings

Established compactness of normalized quasi-convex functions

Proved local and global comparison principles for twisted Monge-Ampère operators

Solved Monge-Ampère equations with arbitrary probability measures

Abstract

We consider degenerate Monge-Amp\`ere equations on compact Hessian manifolds. We establish compactness properties of the set of normalized quasi-convex functions and show local and global comparison principles for twisted Monge-Amp\`ere operators. We then use the Perron method to solve Monge-Amp\`ere equations whose RHS involves an arbitrary probability measure, generalizing works of Cheng-Yau, Delano\"e, Caffarelli-Viaclovsky and Hultgren-\"Onnheim. The intrinsic approach we develop should be useful in deriving similar results on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow conjecture.