Let M be a hyperbolic Riemann surface with the first eigenvalue <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda _1(M)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> denote the distance from a fixed point <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$x_0\in {M}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo>∈</mo> <mi>M</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$r_x$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>r</mi> <mi>x</mi> </msub> </math></EquationSource> </InlineEquation> the injectivity radius at x. We show that there exists a numerical constant <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$c_0>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>c</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> such that if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ r_x\ge c_0 \lambda _1(M)^{-3/4} \rho (x)^{-1/2} $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mi>x</mi> </msub> <mo>≥</mo> <msub> <mi>c</mi> <mn>0</mn> </msub> <msub> <mi>λ</mi> <mn>1</mn> </msub> <msup> <mrow> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>3</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> </msup> <mi>ρ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> holds outside some compact set of M, then the Bergman distance verifies <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ d_\textrm{B}(x,x_0) \gtrsim \log [1+\rho (x)]. $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mtext>B</mtext> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <msub> <mi>x</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>≳</mo> <mo>log</mo> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>+</mo> <mi>ρ</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">]</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation>

An estimate of the Bergman distance on Riemann surfaces

期刊论文
发表日期：

2025年11月8日

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Bo-Yong Chen,
Yuanpu Xiong

摘要

Let M be a hyperbolic Riemann surface with the first eigenvalue $\lambda _1(M)>0$ $λ_{1} (M) > 0$ . Let $\rho $ $ρ$ denote the distance from a fixed point $x_0\in {M}$ $x_{0} \in M$ and $r_x$ $r_{x}$ the injectivity radius at x. We show that there exists a numerical constant $c_0>0$ $c_{0} > 0$ such that if $ r_x\ge c_0 \lambda _1(M)^{-3/4} \rho (x)^{-1/2} $ $r_{x} \geq c_{0} λ_{1} {(M)}^{- 3 / 4} ρ {(x)}^{- 1 / 2}$ holds outside some compact set of M, then the Bergman distance verifies $ d_\textrm{B}(x,x_0) \gtrsim \log [1+\rho (x)]. $ $d_{B} (x, x_{0}) ≳ log [1 + ρ (x)] .$