Green's functions and complex Monge-Ampère equations

2022/02/09 by Guo, Bin, Phong, Duong H., Sturm, Jacob · 4 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2202.04715

Abstract

Uniform L¹ and lower bounds are obtained for the Green's function on compact Kähler 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 Kähler 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-Ampère equations, and is fundamentally non-linear. The lower bounds for the Green's function imply in turn C¹ and C² estimates for complex Monge-Ampère equations with a sharper dependence on the function on the right hand side.

Cited by

• Geometric invariants and the Monge-Ampere equation in Kähler geometry
• Calabi flow with bounded L^p scalar curvature (II)
• Green's function estimates for compact Kähler manifolds and applications
• Estimates of heat kernels and Sobolev-type inequalities for twisted differential forms on compact Kähler manifolds

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