An estimate of the Bergman distance on Riemann surfaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909605881184256 |
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| author | Chen, Bo-Yong Xiong, Yuanpu |
| author_facet | Chen, Bo-Yong Xiong, Yuanpu |
| contents | 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 $d_B(x,x_0) \gtrsim \log [1+ρ(x)]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_05774 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An estimate of the Bergman distance on Riemann surfaces Chen, Bo-Yong Xiong, Yuanpu Complex Variables 32F45, 30F45 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 $d_B(x,x_0) \gtrsim \log [1+ρ(x)]$. |
| title | An estimate of the Bergman distance on Riemann surfaces |
| topic | Complex Variables 32F45, 30F45 |
| url | https://arxiv.org/abs/2505.05774 |