An estimate of the Bergman distance on Riemann surfaces

Bo-Yong Chen; Yuanpu Xiong

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arXiv · 2505.05774

An estimate of the Bergman distance on Riemann surfaces

Bo-Yong Chen·Yuanpu Xiong

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Abstract

Let $M$ be a hyperbolic Riemann surface with the first eigenvalue $λ_1(M)>0$. Let $ρ$ 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 $r_x\ge c_0 λ_1(M)^{-3/4} ρ(x)^{-1/2}$ holds outside some compact set of $M$, then the Bergman distance verifies $d_B(x,x_0) \gtrsim \log [1+ρ(x)]$.

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Bo-Yong Chen, Yuanpu Xiong. 2025-05-09. An estimate of the Bergman distance on Riemann surfaces. https://arxiv.org/abs/2505.05774

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