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Stability of the volume growth rate under quasi-isometries

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Abstract

Kanai proved powerful results on the stability under quasi-isometries of numerous global properties (including the volume growth rate) between non-bordered Riemannian manifolds of bounded geometry. Since his work focuses more on the generality of the spaces considered than on the two-dimensional geometry, Kanai’s hypotheses are not usually satisfied in the context of Riemann surfaces endowed with the Poincaré metric. In this work we try to fill that gap and prove the stability of the volume growth rate by quasi-isometries, under hypotheses that many bordered or non-bordered Riemann surfaces (and even Riemannian surfaces with pinched negative curvature) satisfy. In order to get our results, it is shown that many bordered Riemannian surfaces with pinched negative curvature are bilipschitz equivalent to bordered surfaces with constant negative curvature.

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Acknowledgements

We would like to thank the referees for their careful reading of the manuscript and several useful comments which have helped us to improve the presentation of the paper.

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Authors and Affiliations

  1. Saint Louis University, Madrid Campus Avenida del Valle, 34, 28003, Madrid, Spain

    Ana Granados & Ana Portilla

  2. Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad, 30, 28911, Leganés, Madrid, Spain

    Domingo Pestana & José M. Rodríguez

  3. Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049, Madrid, Spain

    Eva Tourís

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  1. Ana Granados
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  2. Domingo Pestana
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  3. Ana Portilla
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  4. José M. Rodríguez
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  5. Eva Tourís
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Correspondence to Domingo Pestana.

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Supported in part by two grants from Ministerio de Economía y Competititvidad, Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) (MTM2016-78227-C2-1-P and MTM2017-90584-REDT), Spain.

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Granados, A., Pestana, D., Portilla, A. et al. Stability of the volume growth rate under quasi-isometries. Rev Mat Complut 33, 231–270 (2020). https://doi.org/10.1007/s13163-019-00301-6

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  • DOI: https://doi.org/10.1007/s13163-019-00301-6

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