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Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold with Boundary

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Abstract

Let \((M^n,\,g)\) be an n-dimensional compact connected Riemannian manifold with smooth boundary. In this article, we study the effects of the presence of a nontrivial conformal vector field on \((M^n,\,g)\). We used the well-known de-Rham Laplace operator and a nontrivial solution of the famous Fischer–Marsden differential equation to provide two characterizations of the hemisphere \({\mathbb {S}}^n_+(c)\) of constant curvature \(c>0\). As a consequence of the characterization using the Fischer–Marsden equation, we prove the cosmic no-hair conjecture under a given integral condition.

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Acknowledgements

The first named author was partially supported by a grant from CNPq-Brazil. The authors would like to thank Professor Abdênago Barros for fruitful conversations about the results.

Funding

Funding is provided by Conselho Nacional de Desenvolvimento Científico e Tecnológico.

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Authors and Affiliations

  1. Curso de Matemática, Universidade Federal do Delta do Parnaíba- UFDPar, Parnaíba, PI, Brazil

    Israel Evangelista

  2. Universidade Federal do Ceará- UFC, Fortaleza, CE, Brazil

    Antônio Freitas

  3. Instituto Federal de Educação, Ciência e Tecnologia do Ceará (IFCE), Campus Caucaia, Caucaia, CE, Brazil

    Emanuel Viana

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  1. Israel Evangelista
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  2. Antônio Freitas
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  3. Emanuel Viana
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Correspondence to Emanuel Viana.

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Evangelista, I., Freitas, A. & Viana, E. Conformal Vector Fields and the De-Rham Laplacian on a Riemannian Manifold with Boundary. Results Math 77, 217 (2022). https://doi.org/10.1007/s00025-022-01749-7

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