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A Note on Serrin’s Type Problem on Riemannian Manifolds

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Abstract

In this paper, we deal with Serrin-type problems in Riemannian manifolds. First, we obtain a Heintze-Karcher inequality and a Soap Bubble result, with its respective rigidity, when the ambient space has a Ricci tensor bounded below. After, we approach a Serrin problem in bounded domains of manifolds endowed with a closed conformal vector field. Our primary tool, in this case, is a new Pohozaev identity, which depends on the scalar curvature of the manifold. Applications involve Einstein and constant scalar curvature spaces.

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Acknowledgements

The authors would like to thank Alberto Farina and Luciano Mari for their discussions about the object of this paper and several valuable suggestions.

The first author would like to thank the hospitality of the Mathematics Department of Università degli Studi di Torino, where part of this work was carried out (he was supported by CNPq/Brazil Grant 200261/2022-3). The first and third authors have been partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) of the Ministry of Science, Technology and Innovation of Brazil, Grants 316080/2021-7 and 306524/2022-8, and supported by Paraíba State Research Foundation(FAPESQ), Brazil, Grant 3025/2021. The second author is a member of GNAMPA, Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni of INdAM.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal da Paraíba, Cidade Universitária, João Pessoa, Paraíba, 58051-900, Brazil

    Allan Freitas & Márcio Santos

  2. Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, Milano, 20133, Italy

    Alberto Roncoroni

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  1. Allan Freitas
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  2. Alberto Roncoroni
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  3. Márcio Santos
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Correspondence to Márcio Santos.

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Freitas, A., Roncoroni, A. & Santos, M. A Note on Serrin’s Type Problem on Riemannian Manifolds. J Geom Anal 34, 200 (2024). https://doi.org/10.1007/s12220-024-01650-5

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