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Location of geodesics and isoperimetric inequalities in Denjoy domains

Part of: Global differential geometry Riemann surfaces Geometric function theory

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 ([email protected])
José M. Sigarreta
Affiliation:
Facultad de Mateméticas, Universidad Autónoma de Guerrero, Carlos E. Adame No. 54 Col. Garita, 39650 Acalpulco, Guerrero, Mexico ([email protected])
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Abstract

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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

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