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On p-parabolicity of Riemannian manifolds and graphs

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Abstract

Kanai proved that quasi-isometries between Riemannian manifolds with bounded geometry preserve many global properties, including the existence of Green’s function, i.e., non-parabolicity. However, Kanai’s hypotheses are too restrictive. Herein we prove the stability of p-parabolicity (with \(1<p<\infty \)) by quasi-isometries between Riemannian manifolds under weaker assumptions. Also, we obtain some results on the p-parabolicity of graphs and trees; in particular, we characterize p-parabolicity for a large class of trees.

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Acknowledgements

Funding was provided by Ministerio de Economía y Competitividad (ES) (Grant No. PGC2018-098321-B-I00) and Agencia Estatal de Investigación (ES) (Grant No. PID2019-106433GB-I00 / AEI / 10.13039/501100011033).

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Authors and Affiliations

  1. Facultad CC. Sociales de Talavera, Departamento de Análisis Económico y Finanzas, Avda. Real Fábrica de Seda, s/n. 45600 Talavera de la Reina, Toledo, Spain

    Álvaro Martínez-Pérez

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

    José M. Rodríguez

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  1. Álvaro Martínez-Pérez
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  2. José M. Rodríguez
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Correspondence to Álvaro Martínez-Pérez.

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Álvaro Martínez-Pérez supported in part by a Grant from Ministerio de Ciencia, Innovación y Universidades (PGC2018-098321-B-I00), Spain. Second author supported by a Grant from Agencia Estatal de Investigación (PID2019-106433GB-I00 / AEI / 10.13039/501100011033), Spain.

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Martínez-Pérez, Á., Rodríguez, J.M. On p-parabolicity of Riemannian manifolds and graphs. Rev Mat Complut 35, 179–198 (2022). https://doi.org/10.1007/s13163-021-00387-x

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