An estimate of the Bergman distance on Riemann surfaces

TL;DR

This paper establishes a lower bound for the Bergman distance on hyperbolic Riemann surfaces, linking geometric and spectral properties, under specific conditions on injectivity radius and eigenvalues.

Contribution

It provides a new estimate relating the Bergman distance to the hyperbolic distance and spectral data of the surface, under certain geometric conditions.

Findings

Bergman distance grows at least logarithmically with hyperbolic distance.

The estimate depends on the first eigenvalue and injectivity radius.

Conditions involve a lower bound on injectivity radius outside a compact set.

Abstract

Let $M$ be a hyperbolic Riemann surface with the first eigenvalue $\lambda_1(M)>0$. Let $\rho$ denote the distance from a fixed point $x_0\in{M}$ and $r_x$ the injectivity radius at $x$. We show that there exists a numerical constant $c_0>0$ such that if $r_x\ge c_0 \lambda_1(M)^{-3/4} \rho(x)^{-1/2}$ holds outside some compact set of $M$, then the Bergman distance verifies $d_B(x,x_0) \gtrsim \log [1+\rho(x)]$.