Optimal estimate for the gradient of Green functions on degenerating surfaces and applications

TL;DR

This paper establishes a uniform estimate for the gradient of Green functions on degenerating Riemann surfaces, leading to compactness results for certain immersions and surfaces with bounded curvature entropy.

Contribution

It provides a conformally invariant gradient estimate for Green functions on degenerating surfaces, enabling new compactness theorems in geometric analysis.

Findings

Uniform gradient estimate independent of conformal class

Compactness results for immersions with bounded second fundamental form

Results for surfaces with bounded Gaussian curvature entropy

Abstract

In this paper we prove a uniform estimate for the gradient of the Green function on a closed Riemann surface, independent of its conformal class, and we derive compactness results for immersions with L2-bounded second fundamental form and for riemannian surfaces of uniformly bounded gaussian curvature entropy.