Convergence of Riemannian surfaces and convergence of the Bergman kernel

TL;DR

This paper investigates how the Bergman kernel of a sequence of converging Riemannian surfaces behaves, using isoperimetric inequalities to understand the convergence in the Cheeger-Gromov sense.

Contribution

It establishes the convergence properties of the Bergman kernel for sequences of Riemannian surfaces converging in the Cheeger-Gromov sense, linking geometric convergence to kernel behavior.

Findings

Bergman kernels converge under Cheeger-Gromov limits

Isoperimetric inequalities are key to analyzing kernel convergence

Provides new insights into geometric analysis of Riemannian surfaces

Abstract

Let $\{M_j\}$ be a sequence of complete Riemannian surfaces which converges in the sense of Cheeger-Gromov to a complete Riemannian surface $M$. We study the convergence of the Bergman kernel $K_{M_j}$ of $M_j$ by using isoperimetric inequalities.