Estimates of the Bergman kernel on a hyperbolic Riemann surface of finite volume-II

TL;DR

This paper derives off-diagonal and diagonal estimates of the Bergman kernel for tensor powers of the cotangent bundle on finite volume hyperbolic Riemann surfaces, enhancing understanding of its geometric properties.

Contribution

It provides new off-diagonal and diagonal estimates of the Bergman kernel on hyperbolic Riemann surfaces of finite volume, extending previous results.

Findings

Derived off-diagonal estimates for the Bergman kernel

Obtained diagonal estimates using off-diagonal bounds

Enhanced understanding of the kernel's behavior near the injectivity radius

Abstract

In this article, we derive off-diagonal estimates of the Bergman kernel associated to the tensor-powers of the cotangent bundle defined on a hyperbolic Riemann surface of finite volume, when the distance between the points is less than injectivity radius. We then use these estimates to derive estimates of the Bergman kernel along the diagonal.