An estimate of the Bergman distance on Riemann surfaces

TL;DR

This work establishes a quantitative link between Bergman and hyperbolic geometry on noncompact hyperbolic Riemann surfaces with positive first eigenvalue $\lambda_1(M)>0$. The authors develop sharp analytic tools—$L^2$-bar$\partial$ estimates, capacity-Green function relations, Harnack inequalities, and off-diagonal Bergman-kernel bounds—to control the Bergman kernel and the Bergman distance $d_B$ in terms of the hyperbolic distance $\rho(x)$. Under a lower bound on the injectivity radius outside a compact set, namely $r_x \ge c_0 \lambda_1(M)^{-3/4} \rho(x)^{-1/2}$, they prove $d_B(x,x_0) \gtrsim \log[1+\rho(x)]$, providing a concrete bridge between complex-analytic and hyperbolic-geometric growth. The punctured disk example shows the necessity of the radius condition for Bergman completeness, while the approach offers a framework for comparing Bergman and hyperbolic metrics in broad geometric settings."

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 $d_B(x,x_0) \gtrsim \log [1+ρ(x)]$.

Theorems & Definitions (17)

• Theorem 1.1
• Remark 1
• Proposition 2.1
• Lemma 2.2
• proof
• proof : Proof of Proposition \ref{['prop:Harnack']}
• Lemma 3.1
• proof
• Proposition 3.2
• proof