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Mean curvature flow solitons in warped products: nonexistence, rigidity and stability

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Abstract

We deal with several aspects of the geometry of m-dimensional mean curvature flow solitons immersed in a Riemannian warped product \(I\times _{f}M^n\) (\(m\le n\)), with base \(I\subset {\mathbb {R}}\), fiber \(M^n\) and war** function \(f\in C^\infty (I)\). In this context, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space, as well as to obtain nonexistence results concerning these geometric objects. When \(m=n\), we investigate complete two-sided hypersurfaces and, in particular, entire graphs constructed over the fiber \(M^n\) which are mean curvature flow solitons. Furthermore, we infer the stability of closed mean curvature flow solitons with respect to an appropriate stability operator. Applications to self-shrinkers and self-expanders in the Euclidean space and to mean curvature flow solitons in important ambient spaces, like the pseudo-hyperbolic, Schwarzschild and Reissner–Nordström spaces, are also given.

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Funding

The first author is partially supported by CNPq, Brazil, grant 305608/2023-1. The second author is partially supported by Paraíba State Research Foundation (FAPESQ), Brazil, grant 3025/2021, and by CNPq, Brazil, grant 306524/2022-8. The third author is partially supported by CNPq, Brazil, grant 304891/2021-5.

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Authors and Affiliations

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

    Henrique F. de Lima & Marco Antonio L. Velásquez

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

    Márcio S. Santos

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  1. Henrique F. de Lima
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Correspondence to Henrique F. de Lima.

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Lima, H.F.d., Santos, M.S. & Velásquez, M.A.L. Mean curvature flow solitons in warped products: nonexistence, rigidity and stability. Rend. Circ. Mat. Palermo, II. Ser (2024). https://doi.org/10.1007/s12215-024-01066-8

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