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<p>We study reducible spherical conical metrics on compact Riemann surfaces–conformal metrics <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\textrm{d}s^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>d</mtext> <msup> <mi>s</mi> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with finitely many conical singularities and Gaussian curvature <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(K=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, whose monodromy groups are diagonalizable. Our main results are:<UnorderedList Mark="Bullet"> <ItemContent> <p>The existence of reducible spherical conical metrics on a compact Riemann surface is equivalent to the existence of an Abelian differential of the third kind with specified analytic properties.</p> </ItemContent> <ItemContent> <p>Any compact Riemann surface admitting such a metric decomposes into finitely many pieces via suitable geodesic cuts connecting conical singularities and some smooth points. Each piece is isometric to a portion obtained by cutting a football along a geodesic that joins the two conical singularities.</p> </ItemContent> <ItemContent> <p>We give an explicit angle condition that characterizes the existence of such metrics.</p> </ItemContent> </UnorderedList></p>

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Geometric structure and existence of reducible spherical conical metrics

  • 期刊论文

  • 发表日期：

    2026年3月19日

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  • Zhiqiang Wei,
  • Yingyi Wu,
  • Bin Xu

摘要

We study reducible spherical conical metrics on compact Riemann surfaces–conformal metrics \(\textrm{d}s^{2}\) d s 2 with finitely many conical singularities and Gaussian curvature \(K=1\) K = 1 , whose monodromy groups are diagonalizable. Our main results are:

The existence of reducible spherical conical metrics on a compact Riemann surface is equivalent to the existence of an Abelian differential of the third kind with specified analytic properties.

Any compact Riemann surface admitting such a metric decomposes into finitely many pieces via suitable geodesic cuts connecting conical singularities and some smooth points. Each piece is isometric to a portion obtained by cutting a football along a geodesic that joins the two conical singularities.

We give an explicit angle condition that characterizes the existence of such metrics.