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 compact Kähler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an Lq 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.
| Original language | American English |
|---|---|
| Pages (from-to) | 1083-1119 |
| Number of pages | 37 |
| Journal | Journal of Differential Geometry |
| Volume | 127 |
| Issue number | 3 |
| DOIs | |
| State | Published - Jul 2024 |
ASJC Scopus subject areas
- Analysis
- Algebra and Number Theory
- Geometry and Topology