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Stochastically Complete, Parabolic and L1-Liouville Spacelike Submanifolds with Parallel Mean Curvature Vector

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Abstract

We deal with n-dimensional spacelike submanifolds immersed with parallel mean curvature vector h in a pseudo-Riemannian space form \(\mathbb L_{q}^{n+p}(c)\) of index 1 ≤ q ≤ p and constant sectional curvature c ∈{− 1,0,1}. Considering the cases when h is either spacelike or timelike, we are able to prove that such a spacelike submanifold is either totally umbilical or it holds a lower estimate for the supremum of the norm of its traceless second fundamental form, occurring equality if the spacelike submanifold is pseudo-umbilical and its principal curvatures are constant. In our approach, we use three main core concepts: Stochastic completeness, parabolicity and L1-Liouville property.

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Acknowledgements

The authors would like to thank the referee for his/her valuable suggestions and comments which enabled them to improve this paper. The authors would also like to thank professor Fábio Reis dos Santos for fruitful discussions concerning the thematic of this paper. The first author is partially supported by CAPES, Brazil. The second and third authors are partially supported by CNPq, Brazil, grants 301970/2019-0 and 311224/2018-0, respectively.

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

    Weiller F. C. Barboza, Henrique F. de Lima & Marco Antonio L. Velásquez

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  1. Weiller F. C. Barboza
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  2. Henrique F. de Lima
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  3. Marco Antonio L. Velásquez
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Correspondence to Henrique F. de Lima.

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Barboza, W.F.C., de Lima, H.F. & Velásquez, M.A.L. Stochastically Complete, Parabolic and L1-Liouville Spacelike Submanifolds with Parallel Mean Curvature Vector. Potential Anal 60, 27–43 (2024). https://doi.org/10.1007/s11118-022-10043-8

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