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Yamabe problem in the presence of singular Riemannian Foliations

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Abstract

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

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Acknowledgements

We thank Monica Clapp for useful conversations about the question posted in the introduction. We also thank Jimmy Petean for useful comments.

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Author notes

  1. Diego Corro

    Present address: Fakultät für Mathematik, Karlsruher Institute für Technologie, Karlsruhe, Germany

Authors and Affiliations

  1. Mathematics Institute of the National Autonomous University of Mexico (UNAM), Oaxaca, Mexico

    Diego Corro

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM), Mexico City, Mexico

    Juan Carlos Fernandez

  3. Investigadora por México at the Mathematics Institute of the National Autonomous University of Mexico (UNAM), Oaxaca, Mexico

    Raquel Perales

Authors
  1. Diego Corro
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  2. Juan Carlos Fernandez
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  3. Raquel Perales
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Correspondence to Juan Carlos Fernandez.

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Communicated by A. Mondino.

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D. Corro: Supported by DGAPA-Fellowship associated to the Mathematics Institute of UNAM, campus Oaxaca, and by DFG-Eigenestelle Fellowship CO 2359/1-1. J. C. Fernandez: Partially supported by Professor Christina Sormani’s NSF Reaserch Grant DMS-1612049.

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Corro, D., Fernandez, J.C. & Perales, R. Yamabe problem in the presence of singular Riemannian Foliations. Calc. Var. 62, 26 (2023). https://doi.org/10.1007/s00526-022-02359-5

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