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arXiv · 2106.14740

Monge-Amp\`ere equations on compact Hessian manifolds

Vincent Guedj·Tat Dat Tô
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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.

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Vincent Guedj, Tat Dat Tô. 2021-06-28. Monge-Amp\`ere equations on compact Hessian manifolds. https://arxiv.org/abs/2106.14740

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