Skip to main content
SpringerLink
Log in

The topology of balls in Riemannian surfaces and Gromov hyperbolicity

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Alvarez, V., Pestana, D., Rodríguez, J.M.: Isoperimetric inequalities in Riemann surfaces of infinite type. Rev. Mat. Iberoamericana 15, 353–427 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alvarez, V., Portilla, A., Rodríguez, J.M., Tourís, E.: Gromov hyperbolicity of Denjoy domains. Geom. Dedicata 121, 221–245 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alvarez, V., Rodríguez, J.M., Yakubovich, V.A.: Subadditivity of p-harmonic “measure” on graphs. Michigan Math. J. 49, 47–64 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Anderson, J.W.: Hyperbolic Geometry. Springer, London (1999)

    Book  MATH  Google Scholar 

  5. Balogh, Z.M., Buckley, S.M.: Geometric characterizations of Gromov hyperbolicity. Invent. Math. 153, 261–301 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Beardon, A.F., Pommerenke, Ch.: The Poincaré metric of a plane domain. J. London Math. Soc. 18, 475–483 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benoist, Y.: Convexes hyperboliques et fonctions quasisymétriques. Publ. Math. Inst. Hautes Études Sci. 97, 181–237 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bonk, M., Heinonen, J., Koskela, P.: Uniformizing Gromov hyperbolic spaces. Astérisque No. 270 (2001)

  9. Buser, P.: Geometry and Spectra of Compact Riemann Surfaces. Birkhäuser, Boston (1992)

    MATH  Google Scholar 

  10. Cantón, A., Fernández, J.L., Pestana, D., Rodríguez, J.M.: On harmonic functions on trees. Potential Anal. 15, 199–244 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carleson, L.: An interpolation problem for bounded analytic functions. Am. J. Math. 80, 921–930 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chavel, I., Feldman, E.A.: Cylinders on surfaces. Comment. Math. Helv. 53, 439–447 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  13. Coornaert, M., Delzant, T., Papadopoulos, A.: Notes sur les groups hyperboliques de Gromov. IRMA, Strasbourg (1989)

    Google Scholar 

  14. Fernández, J.L., Rodríguez, J.M.: The exponent of convergence of Riemann surfaces, Bass Riemann surfaces. Ann. Acad. Sci. Fenn. Series AI 15, 165–183 (1990)

    MATH  Google Scholar 

  15. Fernández, J.L., Rodríguez, J.M.: Area growth and Green’s function of Riemann surfaces. Arkiv för matematik 30, 83–92 (1992)

    Article  MATH  Google Scholar 

  16. Ghys, E., de la Harpe, P.: Sur les Groupes Hyperboliques d’après Mikhael Gromov. Progress in Mathematics, vol. 83. Birkhäuser (1990)

  17. Gonzalo, J.: A radius estimate for doubly connected balls in negatively curved surfaces. (Preprint)

  18. Gromov, M.: Curvature, diameter and Betti numbers. Comment Math. Hlv. 56, 179–195 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in group theory MSRI Publ 8, pp. 75–263. Springer, Berlin (1987)

    Chapter  Google Scholar 

  20. Grove, K., Petersen, P.: Bounding homotopy types by geometry. Ann. Math. 128, 195–206 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  21. Grove, K., Petersen, P., Wu, J.Y.: Geometric finiteness theorems via controled topology. Invent. Math. 99, 205–213 (1990). Correction in Invent. Math. 104, 221–222 (1991)

    Google Scholar 

  22. Hästö, P.A.: Gromov hyperbolicity of the \(j_G\) and \(\widetilde{j}_G\) metrics. Proc. Am. Math. Soc. 134, 1137–1142 (2006)

    Article  MATH  Google Scholar 

  23. Hästö, P.A., Lindén, H., Portilla, A., Rodríguez, J.M., Tourís, E.: Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics. J. Math. Soc. Japan 64, 245–259 (2012)

    Article  Google Scholar 

  24. Hästö, P., Portilla, A., Rodríguez, J.M., Tourís, E.: Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains. B. London Math. Soc. 42, 282–294 (2010)

    Article  MATH  Google Scholar 

  25. Hästö, P., Portilla, A., Rodríguez, J.M., Tourís, E.: Uniformly separated sets and Gromov hyperbolicity of domains with the quasihyperbolicity metric. Mediterr. J. Math. 8, 47–65 (2011)

    Article  Google Scholar 

  26. Holopainen, I., Soardi, P.M.: \(p\)-Harmonic functions on graphs and manifolds. Manuscripta Math. 94, 95–110 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kanai, M.: Rough isometries and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan 37, 391–413 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  28. Karlsson, A., Noskov, G.A.: The Hilbert metric and Gromov hyperbolicity. Enseign. Math. 48, 73–89 (2002)

    MathSciNet  MATH  Google Scholar 

  29. Kobayashi, S.: On conjugate and cut loci. Studies in Global Geometry and Analysis. Math. Assoc. Am. pp. 96–122 (1967)

  30. Myers, S.B.: Connections between Differrential Geometry and Topology: I. Simply Connected Surfaces. Duke Math. J. 1, 376–391 (1935)

    Article  MathSciNet  Google Scholar 

  31. Myers, S.B.: Connections between Differrential Geometry and Topology: II. Closed Surfaces. Duke Math. J. 2, 95–102 (1936)

    Article  MathSciNet  Google Scholar 

  32. Ortega, J., Seip, K.: Harmonic measure and uniform densities. Indiana Univ. Math. J. 53, 905–923 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Petersen, P.: Riemannian geometry. Graduate Texts in Mathematics, 171. Springer, New York (2006)

    Google Scholar 

  34. Portilla, A., Rodríguez, J.M., Tourís, E.: Gromov hyperbolicity through decomposition of metric spaces II. J. Geom. Anal. 14, 123–149 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  35. Portilla, A., Rodríguez, J.M., Tourís, E.: The topology of balls and Gromov hyperbolicity of Riemann surfaces. Diff. Geom. Appl. 21, 317–335 (2004)

    Article  MATH  Google Scholar 

  36. Portilla, A., Rodríguez, J.M., Tourís, E.: The role of funnels and punctures in the Gromov hyperbolicity of Riemann surfaces. Proc. Edinburgh Math. Soc. 49, 399–425 (2006)

    Article  MATH  Google Scholar 

  37. Portilla, A., Tourís, E.: A new characterization of Gromov hyperbolicity for non-constant negatively curved surfaces. Publ. Matem. 53, 83–110 (2009)

    Article  MATH  Google Scholar 

  38. Randol, B.: Cylinders in Riemann surfaces. Comment. Math. Helv. 54, 1–5 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  39. Ratcliffe, J.G.: Foundations of Hyperbolic Manifolds. Springer, New York (1994)

    Book  MATH  Google Scholar 

  40. Rodríguez, J.M., Tourís, E.: Gromov hyperbolicity of Riemann surfaces. Acta Math. Sinica 23, 209–228 (2007)

    Article  MATH  Google Scholar 

  41. Soardi, P.M.: Rough isometries and Dirichlet finite harmonic functions on graphs. Proc. Am. Math. Soc. 119, 1239–1248 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

J. Gonzalo Partially supported by grants MTM2007-61982 from MEC Spain and MTM2008-02686 from MICINN Spain. A. Portilla and E. Tourís supported in part by a grant MTM 2009-12740-C03-01 from MICINN Spain. A. Portilla, J. M. Rodrígurz and E. Tourís are supported in part by two grants MTM 2009-07800 and MTM 2008-02829-E from MICINN Spain. J. M. Rodríguez supported in part by a grant from CONACYT (CONACYT-UAG I0110/62/10), México.

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 , Madrid, Spain

    Jesús Gonzalo & Eva Tourís

  2. St. Louis University (Madrid Campus), Avenida del Valle 34, 28003 , Madrid, Spain

    Ana Portilla

  3. Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés (Madrid), Spain

    José M. Rodríguez

Authors
  1. Jesús Gonzalo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ana Portilla
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José M. Rodríguez
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eva Tourís
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ana Portilla.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gonzalo, J., Portilla, A., Rodríguez, J.M. et al. The topology of balls in Riemannian surfaces and Gromov hyperbolicity. Math. Z. 275, 741–760 (2013). https://doi.org/10.1007/s00209-013-1158-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-013-1158-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation