Monge-Amp\`ere equations on compact Hessian manifolds
Differential Geometry
2021-06-29 v1 Analysis of PDEs
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.
Cite
@article{arxiv.2106.14740,
title = {Monge-Amp\`ere equations on compact Hessian manifolds},
author = {Vincent Guedj and Tat Dat Tô},
journal= {arXiv preprint arXiv:2106.14740},
year = {2021}
}