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arXiv · 2202.04715

Green's functions and complex Monge-Amp\`ere equations

Bin Guo·Duong H. Phong·Jacob Sturm
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Abstract

Uniform $L^1$ and lower bounds are obtained for the Green's function on compact K\"ahler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only on integral bounds for the volume form and certain of its derivatives. In particular, a uniform lower bound for the Green's function on K\"ahler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an $L^q$ norm for the volume form for some $q>1$. The proof relies on auxiliary Monge-Amp\`ere equations, and is fundamentally non-linear. The lower bounds for the Green's function imply in turn $C^1$ and $C^2$ estimates for complex Monge-Amp\`ere equations with a sharper dependence on the function on the right hand side.

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Bin Guo, Duong H. Phong, Jacob Sturm. 2022-02-09. Green's functions and complex Monge-Amp\`ere equations. https://arxiv.org/abs/2202.04715

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