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<p>We show that every open Riemann surface <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>X</mi> </math></EquationSource> <EquationSource Format="TEX">$X$</EquationSource> </InlineEquation> can be obtained by glueing together a countable collection of equilateral triangles, in such a way that every vertex belongs to finitely many triangles. Equivalently, <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>X</mi> </math></EquationSource> <EquationSource Format="TEX">$X$</EquationSource> </InlineEquation> is a <i>Belyi surface</i>: There exists a holomorphic branched covering <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mover accent="true"> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">ˆ</mo> </mover> </math></EquationSource> <EquationSource Format="TEX">$f\colon X\to \hat{\mathbb{C}}$</EquationSource> </InlineEquation> that is branched only over −1, 1 and <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">∞</mi> </math></EquationSource> <EquationSource Format="TEX">$\infty $</EquationSource> </InlineEquation>. It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.</p>

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Non-compact Riemann surfaces are equilaterally triangulable

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月25日

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  • Christopher J. Bishop,
  • Lasse Rempe

摘要

We show that every open Riemann surface X $X$ can be obtained by glueing together a countable collection of equilateral triangles, in such a way that every vertex belongs to finitely many triangles. Equivalently, X $X$ is a Belyi surface: There exists a holomorphic branched covering f : X → C ˆ $f\colon X\to \hat{\mathbb{C}}$ that is branched only over −1, 1 and ∞ $\infty $ . It follows that every Riemann surface is a branched cover of the sphere, branched only over finitely many points.