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<p>The possible omega limit sets of simple geodesics for meromorphic connections on compact Riemann surfaces have been studied by Abate, Tovena and Bianchi. In this paper we study the same problem for infinite self-intersecting geodesics. In the first part of the paper we study relation among meromorphic <i>k</i>-differentials, singular flat metrics and meromorphic connections. In the second part we define the notion of meromorphic <i>G</i>-differential for a multiplicative group <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2190_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="56" /> </InlineMediaObject> <EquationSource Format="TEX">\(G\subset \mathbb {C}^*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mo>∗</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> and we show a relation between meromorphic <i>G</i>-differentials and meromorphic connections. Moreover we prove a Poincaré-Bendixson theorem for infinite self-intersecting geodesics of meromorphic connections monodromy in <i>G</i>, with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2190_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">\(\arg G^k=\{0\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>arg</mo> <msup> <mi>G</mi> <mi>k</mi> </msup> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mn>0</mn> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for some <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2190_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(k\in \mathbb {N}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Flat structure of meromorphic connections on Riemann surfaces

  • 期刊论文

  • 发表日期：

    2025年9月23日

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  • Karim Rakhimov

摘要

The possible omega limit sets of simple geodesics for meromorphic connections on compact Riemann surfaces have been studied by Abate, Tovena and Bianchi. In this paper we study the same problem for infinite self-intersecting geodesics. In the first part of the paper we study relation among meromorphic k-differentials, singular flat metrics and meromorphic connections. In the second part we define the notion of meromorphic G-differential for a multiplicative group \(G\subset \mathbb {C}^*\) G ⊂ C ∗ and we show a relation between meromorphic G-differentials and meromorphic connections. Moreover we prove a Poincaré-Bendixson theorem for infinite self-intersecting geodesics of meromorphic connections monodromy in G, with \(\arg G^k=\{0\}\) arg G k = { 0 } for some \(k\in \mathbb {N}\) k ∈ N .