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Nonexistence and rigidity of spacelike mean curvature flow solitons immersed in a GRW spacetime

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Abstract

We study the nonexistence and rigidity of an important class of particular cases of trapped submanifolds, more precisely, n-dimensional spacelike mean curvature flow solitons related to the closed conformal timelike vector field \(\mathcal K=f(t)\partial _t\) (\(t\in I\subset \mathbb R\)) which is globally defined on an \((n+p+1)\)-dimensional generalized Robertson–Walker (GRW) spacetime \(-I\times _fM^{n+p}\) with warping function \(f\in C^\infty (I)\) and Riemannian fiber \(M^{n+p}\), via applications of suitable generalized maximum principles and under certain constraints on f and on the curvatures of \(M^{n+p}\). In codimension 1, we also obtain new Calabi–Bernstein-type results concerning the spacelike mean curvature flow soliton equation in a GRW spacetime.

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Acknowledgements

The authors would like to thank the referee for his/her valuable suggestions and useful comments to improve the paper. The first and second 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 301970/2019-0, respectively. The first, third and fourth authors are partially supported by Paraíba State Research Foundation (FAPESQ), Brazil, Grant 3025/2021. The first author has also been partially funded by the public call n. 03 Produtividade em Pesquisa PROPESQ/PRPG/UFPB, proposal code PIA13495-2020, and Programa Primeiros Projetos Grant 2021/3175, Paraíba State Research Foundation (FAPESQ).

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal da Paraíba, João Pessoa, Paraíba, 58.051-900, Brazil

    Allan Freitas, Márcio S. Santos & Joyce S. Sindeaux

  2. Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, Paraíba, 58.429-970, Brazil

    Henrique F. de Lima

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  1. Allan Freitas
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  2. Henrique F. de Lima
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  4. Joyce S. Sindeaux
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Correspondence to Allan Freitas.

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Freitas, A., de Lima, H.F., Santos, M.S. et al. Nonexistence and rigidity of spacelike mean curvature flow solitons immersed in a GRW spacetime. Ann Glob Anal Geom 63, 2 (2023). https://doi.org/10.1007/s10455-022-09879-5

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