Isoperimetric inequalities in Riemann surfaces of infinite type

Álvarez, Venancio; Pestana, Domingo; Rodríguez, José M.

doi:10.4171/rmi/260

Cited by 29 publications

(35 citation statements)

References 1 publication

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“…The uniformly separated sets play a central role in the study of hyperbolic isoperimetric inequalities in open Riemann surfaces (see [2,Theorem 1] and [12, Theorems 3 and 4]), and in other topics in complex analysis, such as harmonic measure (see [21]). Interesting relationships exist between the hyperbolic isoperimetric inequality and other conformal invariants of a Riemann surface (see, for example, [2], [10, p. 95], [12] and [31, p. 333…”

Section: Remarks 510mentioning

confidence: 99%

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The Role of Funnels and Punctures in the Gromov Hyperbolicity of Riemann Surfaces

Portilla¹,

Rodríguez²,

Tourís³

2006

Proceedings of the Edinburgh Mathematical Society

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We prove results on geodesic metric spaces which guarantee that some spaces are not hyperbolic in the Gromov sense. We use these theorems in order to study the hyperbolicity of Riemann surfaces. We obtain a criterion on the genus of a surface which implies non-hyperbolicity. We also include a characterization of the hyperbolicity of a Riemann surface S * obtained by deleting a closed set from one original surface S. In the particular case when the closed set is a union of continua and isolated points, the results clarify the role of punctures and funnels (and other more general ends) in the hyperbolicity of Riemann surfaces.

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Section: Remarks 510mentioning

confidence: 99%

“…Interesting relationships exist between the hyperbolic isoperimetric inequality and other conformal invariants of a Riemann surface (see, for example, [2], [10, p. 95], [12] and [31, p. 333…”

Section: Remarks 510mentioning

confidence: 99%

The Role of Funnels and Punctures in the Gromov Hyperbolicity of Riemann Surfaces

Portilla¹,

Rodríguez²,

Tourís³

2006

Proceedings of the Edinburgh Mathematical Society

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“…, r} e 2s − 1 δ D * n (z), for every z ∈ V * n , and this gives the first inequality in (2). Mediterr.…”

Section: Vol 8 (2011)mentioning

confidence: 99%

Uniformly Separated Sets and Gromov Hyperbolicity of Domains with the Quasihyperbolic Metric

et al. 2010

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In this article we prove comparative results on the Gromov hyperbolicity of plane domains equipped with the quasihyperbolic metric. By a comparative result we mean one which assumes hyperbolicity in one domain and obtains it in a different domain somehow related to the original one. We derive a characterization (simple to check in practical cases) of the Gromov hyperbolicity of a plane domain Ω * obtained by deleting from the original domain Ω any uniformly separated union of compact sets. We present as well a result about stability of hyperbolicity.Mathematics Subject Classification (2010). Primary 30F45; Secondary 53C23, 30C99.

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“…This kind of surfaces are becoming more and more important in Geometric Theory of Functions, since, on the one hand, they are a very general type of Riemann surfaces, and, on the other hand, they are more manageable due to its symmetry. For instance, Garnett and Jones have proved in [15] the Corona Theorem for Denjoy domains, and in [2], [42] the authors have got characterizations of Denjoy domains which satisfy a linear isoperimetric inequality.…”

Section: Introductionmentioning

confidence: 99%

A Very Simple Characterization of Gromov Hyperbolicity for a Special Kind of Denjoy Domains

Portilla¹,

Rodrı́guez²,

Tourís³

2011

Journal of the Korean Mathematical Society

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Abstract. In this paper we provide characterizations for the Gromov hyperbolicity of some particular Denjoy domains and besides some sufficient conditions to guarantee or discard the hyperbolicity of some others. The conditions obtained involve just the lengths of some special simple closed geodesics in the domain. These results, on the one hand, show the extraordinary complexity of determining the hyperbolicity of a domain and, on the other hand, allow us to construct easily a large variety of both hyperbolic and non-hyperbolic domains.

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Isoperimetric inequalities in Riemann surfaces of infinite type

Cited by 29 publications

References 1 publication

The Role of Funnels and Punctures in the Gromov Hyperbolicity of Riemann Surfaces

The Role of Funnels and Punctures in the Gromov Hyperbolicity of Riemann Surfaces

Uniformly Separated Sets and Gromov Hyperbolicity of Domains with the Quasihyperbolic Metric

A Very Simple Characterization of Gromov Hyperbolicity for a Special Kind of Denjoy Domains

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