An estimate of the Bergman distance on Riemann surfaces

2025/05/09 by Bo-Yong Chen, Chen, Bo-Yong, Yuanpu Xiong +1
Mathematics · #30F45 #32F45 #Complex Variables (math.CV) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2505.05774

openalex publication_date 2025/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a hyperbolic Riemann surface with the first eigenvalue λ₁(M)>0. Let ρ denote the distance from a fixed point x₀∈M and r_x the injectivity radius at x. We show that there exists a numerical constant c₀>0 such that if r_x≥ c₀ λ₁(M)^-3/4 ρ(x)^-1/2 holds outside some compact set of M, then the Bergman distance verifies d_B(x,x₀) \gtrsim log [1+ρ(x)].

Citations

• Capacities, Green function and Bergman functions

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