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A Poincaré–Bendixson theorem for meromorphic connections on Riemann surfaces

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Abstract

We shall prove a Poincaré–Bendixson theorem describing the asymptotic behavior of geodesics for a meromorphic connection on a compact Riemann surface. We shall also briefly discuss the case of non-compact Riemann surfaces, and study in detail the geodesics for a holomorphic connection on a complex torus.

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Acknowledgments

Both authors were partially supported by the FIRB 2012 project Differential geometry and geometric function theory. The first author has also been partially supported by GNSAGA (INdAM).

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Pontecorvo 5, 56127, Pisa, Italy

    Marco Abate & Fabrizio Bianchi

  2. Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route de Narbonne, 31062, Toulouse, France

    Fabrizio Bianchi

Authors
  1. Marco Abate
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  2. Fabrizio Bianchi
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Correspondence to Marco Abate.

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Abate, M., Bianchi, F. A Poincaré–Bendixson theorem for meromorphic connections on Riemann surfaces. Math. Z. 282, 247–272 (2016). https://doi.org/10.1007/s00209-015-1540-6

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  • DOI: https://doi.org/10.1007/s00209-015-1540-6

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