The topology of balls in Riemannian surfaces and Gromov hyperbolicity

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• 作者：Jesús Gonzalo (1)
  Ana Portilla (2)
  José M. Rodríguez (3)
  Eva Tourís (1)
  
• 关键词：Metric balls ; Topology of balls ; Riemannian surfaces ; Gromov hyperbolicity ; Riemann surfaces ; Poincaré metric ; 53C20 ; 53C21 ; 30F20 ; 30F45
• 刊名：Mathematische Zeitschrift
• 出版年：2013
• 出版时间：December 2013
• 年：2013
• 卷：275
• 期：3-4
• 页码：741-760
• 全文大小：
• 作者单位：Jesús Gonzalo (1)
  Ana Portilla (2)
  José M. Rodríguez (3)
  Eva Tourís (1)
  
  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049, Madrid, Spain
  2. St. Louis University (Madrid Campus), Avenida del Valle 34, 28003, Madrid, Spain
  3. Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés (Madrid), Spain
  
• ISSN：1432-1823

文摘

For each $k>0$ we find an explicit function $f_k$ such that the topology of $S$ inside the ball $B_S(p,r)$ is ‘bounded’ by $f_k(r)$ for every complete Riemannian surface (compact or non-compact) $S$ with $K \ge -k^2$ , every $p \in S$ and every $r>0$ . Using this result, we obtain a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a Riemann surface $S^*$ (with its own Poincaré metric) obtained by deleting from one original surface $S$ any uniformly separated union of continua and isolated points.