Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T09:25:52.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Location of geodesics and isoperimetric inequalities in Denjoy domains

Part of: Riemann surfaces Geometric function theory Global differential geometry

Published online by Cambridge University Press:  15 June 2011

José M. Rodríguez  and
José M. Sigarreta
José M. Rodríguez
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Madrid, Spain (jomaro@math.uc3m.es)
José M. Sigarreta
Affiliation:
Facultad de Mateméticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54 Col. Garita, 39650 Acalpulco, Guerrero, Mexico (josemariasigarretaalmira@yahoo.es)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We find approximate solutions (chord–arc curves) for the system of equations of geodesics in Ω∩ℍ for every Denjoy domain Ω, with respect to both the Poincaré and the quasi-hyperbolic metrics. We also prove that these chord–arc curves are uniformly close to the geodesics. As an application of these results, we obtain good estimates for the lengths of simple closed geodesics in any Denjoy domain, and we improve the characterization in a 1999 work by Alvarez et al. on Denjoy domains satisfying the linear isoperimetric inequality.

Keywords

geodesicquasi-geodesicchord–arcPoincaré metriclinear isoperimetric inequalityDenjoy domain

MSC classification

Secondary: 30F45: Conformal metrics (hyperbolic, Poincaré, distance functions) 53C22: Geodesics 30C99: None of the above, but in this section
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 245 - 269
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Aikawa, H., Positive harmonic functions of finite order in a Denjoy type domain, Proc. Am. Math. Soc. 131 (2003), 3873–3881.CrossRefGoogle Scholar
2.Alvarez, V., Pestana, D. and Rodríguez, J. M., Isoperimetric inequalities in Riemann surfaces of infinite type, Rev. Mat. Ibero. 15 (1999), 353–427.CrossRefGoogle Scholar
3.Alvarez, V., Portilla, A., Rodríguez, J. M. and Tourís, E., Gromov hyperbolicity of Denjoy domains, Geom. Dedicata 121 (2006), 221–245.CrossRefGoogle Scholar
4.Anderson, J. W., Hyperbolic geometry (Springer, 1999).CrossRefGoogle Scholar
5.Balogh, Z. M. and Buckley, S. M., Geometric characterizations of Gromov hyperbolicity, Invent. Math. 153 (2003), 261–301.CrossRefGoogle Scholar
6.Beardon, A. F. and Pommerenke, Ch., The Poincaré metric of a plane domain, J. Lond. Math. Soc. 18 (1978), 475–483.CrossRefGoogle Scholar
7.Bonk, M., Quasi-geodesics segments and Gromov hyperbolic spaces, Geom. Dedicata 62 (1996), 281–298.CrossRefGoogle Scholar
8.Bonk, M., Heinonen, J. and Koskela, P., Uniformizing Gromov hyperbolic spaces, Astérisque 270 (2001), 1–99.Google Scholar
9.Bonk, M. and Schramm, O., Embeddings of Gromov hyperbolic spaces, Geom. Funct. Analysis 10 (2000), 266–306.CrossRefGoogle Scholar
10.Chavel, I., Eigenvalues in Riemannian geometry (Academic Press, New York, 1984).Google Scholar
11.Chazarain, J., Spectre des opérateurs elliptiques et flots hamiltoniens, in Séminaire Bourbaki 1974–75, Lecture Notes in Mathematics, Volume 514 (Springer, 1976).Google Scholar
12.Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, in Problems in analysis, ed. Gunning, R. C., pp. 195–199 (Princeton University Press, 1970).Google Scholar
13.de Verdiere, Y. Colin, Quasi-modos sur les variétés Riemanniennes, Invent. Math. 43 (1977), 15–52.CrossRefGoogle Scholar
14.Fernández, J. L. and Rodríguez, J. M, The exponent of convergence of Riemann surfaces: Bass Riemann surfaces, Annales Acad. Sci. Fenn. Math. 15 (1990), 165–183.CrossRefGoogle Scholar
15.Garnett, J. and Jones, P., The Corona Theorem for Denjoy domains, Acta Math. 155 (1985), 27–40.CrossRefGoogle Scholar
16.Ghys, E. and de la Harpe, P., Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Mathematics, Volume 83 (Birkhäuser, 1990).CrossRefGoogle Scholar
17.González, M. J., An estimate on the distortion of the logarithmic capacity, Proc. Am. Math. Soc. 126 (1998), 1429–1431.CrossRefGoogle Scholar
18.Guillemin, V., Lectures on spectral theory of elliptic operators, Duke Math. J. 44 (1977), 485–517.CrossRefGoogle Scholar
19.Hästö, P. A., Gromov hyperbolicity of the jG and metrics, Proc. Am. Math. Soc. 134 (2006), 1137–1142.CrossRefGoogle Scholar
20.Hästö, P. A., Lindén, H., Portilla, A., Rodríguez, J. M. and Tourís, E., Gromov hyperbolicity of Denjoy domains with hyperbolic and quasihyperbolic metrics, J. Math. Soc. Jpn, in press.Google Scholar
21.Hästö, P. A., Portilla, A., Rodríguez, J. M. and Tourís, E., Comparative Gromov hyperbolicity results for the hyperbolic and quasihyperbolic metrics, Complex Variables 55 (2010), 127–135.CrossRefGoogle Scholar
22.Hästö, P. A., Portilla, A., Rodríguez, J. M. and Tourís, E., Gromov hyperbolic equivalence of the hyperbolic and quasihyperbolic metrics in Denjoy domains, Bull. Lond. Math. Soc. 42 (2010), 282–294.CrossRefGoogle Scholar
23.Hästö, P. A., Portilla, A., Rodríguez, J. M. and Tourís, E., Uniformly separated sets and Gromov hyperbolicity of domains with the quasihyperbolic metric, Medit. J. Math. 8 (2011), 47–65.Google Scholar
24.Karlsson, A. and Noskov, G. A., The Hilbert metric and Gromov hyperbolicity, Enseign. Math. 48 (2002), 73–89.Google Scholar
25.Lindén, H., Gromov hyperbolicity of certain conformal invariant metrics, Annales Acad. Sci. Fenn. Math. 32 (2007), 279–288.Google Scholar
26.Minda, D., A reflection principle for the hyperbolic metric and applications to geometric function theory, Complex Variables 8 (1987), 129–144.Google Scholar
27.Nicholls, P. J., The ergodic theory of discrete groups, Lecture Notes Series, Volume 143 (Cambridge University Press, 1989).CrossRefGoogle Scholar
28.Portilla, A., Rodríguez, J. M. and Tourís, E., Gromov hyperbolicity through decomposition of metric spaces, II, J. Geom. Analysis 14 (2004), 123–149.CrossRefGoogle Scholar
29.Portilla, A., Rodríguez, J. M. and Tourís, E., The topology of balls and Gromov hyperbolicity of Riemann surfaces, Diff. Geom. Applic. 21 (2004), 317–335.CrossRefGoogle Scholar
30.Portilla, A., Rodríguez, J. M. and Tourís, E., The role of funnels and punctures in the Gromov hyperbolicity of Riemann surfaces, Proc. Edinb. Math. Soc. 49 (2006), 399–425.CrossRefGoogle Scholar
31.Portilla, A., Rodríguez, J. M. and Tourís, E., A real variable characterization of Gromov hyperbolicity of flute surfaces, Osaka J. Math. 48 (2011), 179–207.Google Scholar
32.Portilla, A. and Tourís, E., A characterization of Gromov hyperbolicity of surfaces with variable negative curvature, Publ. Mat. 53 (2009), 83–110.CrossRefGoogle Scholar
33.Rodríguez, J. M. and Tourís, E., Gromov hyperbolicity through decomposition of metric spaces, Acta Math. Hungar. 103 (2004), 53–84.CrossRefGoogle Scholar
34.Rodríguez, J. M. and Tourís, E., A new characterization of Gromov hyperbolicity for Riemann surfaces, Publ. Mat. 50 (2006), 249–278.CrossRefGoogle Scholar
35.Rodríguez, J. M. and Tourís, E., Gromov hyperbolicity of Riemann surfaces, Acta Math. Sinica 23 (2007), 209–228.CrossRefGoogle Scholar
36.Sullivan, D., Related aspects of positivity in Riemannian geometry, J. Diff. Geom. 25 (1987), 327–351.Google Scholar
37.Tourís, E., Graphs and Gromov hyperbolicity of non-constant negatively curved surfaces, J. Math. Analysis Applic., in press.Google Scholar
38.Väisälä, J., Hyperbolic and uniform domains in Banach spaces, Annales Acad. Sci. Fenn. Math. 30 (2005), 261–302.Google Scholar
39.Väisälä, J., Gromov hyperbolic spaces, Expo. Math. 23 (2005), 187–231.CrossRefGoogle Scholar