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Spacelike translating solitons of mean curvature flow with forcing term in Lorentzian product spaces

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Abstract

In this paper we study translating solitons with a forcing term immersed in a Lorentzian product space \(\mathbb R_1\times {\mathbb {P}}^n\). Under suitable curvature constraints on the curvatures of the Riemannian base \({\mathbb {P}}^n\) we deduce an Omori-Yau maximum principle, and as applications we get a nonexistence result. Besides, we also prove that any entire spacelike downward translating graph that never intersects null infinity is mean convex. Furthermore, under suitable assumption on the base \({\mathbb {P}}^n\), we prove a boundedness of the second fundamental form of such translating solitons.

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Acknowledgements

We would like to thank the referee for his carefull reading and many useful comments which improved the presentation of this paper.

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  1. CPMAT-IM, Universidade Federal de Alagoas, Maceió, Alagoas, 57072-900, Brazil

    Márcio Batista & Pedro Carvalho

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  1. Márcio Batista
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  2. Pedro Carvalho
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Correspondence to Márcio Batista.

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This work was partially supported by Alagoas Research Foundation (FAPEAL), Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) (grant: 308440/2021-8 and 405468/2021-0 to M.B.) and Coordenação de Aperfeiçoamento de Pessoal de nível Superior (CAPES) (grant: 001 to both authors), Brazil.

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Batista, M., Carvalho, P. Spacelike translating solitons of mean curvature flow with forcing term in Lorentzian product spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 114 (2023). https://doi.org/10.1007/s13398-023-01445-3

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