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ON THE GEOMETRY OF SPACELIKE MEAN CURVATURE FLOW SOLITONS IMMERSED IN A GRW SPACETIME

Part of: Global differential geometry

Published online by Cambridge University Press:  15 September 2023

HENRIQUE F. DE LIMA Open the ORCID record for HENRIQUE F. DE LIMA [Opens in a new window] ,
WALLACE F. GOMES ,
MÁRCIO S. SANTOS  and
MARCO ANTONIO L. VELÁSQUEZ Open the ORCID record for MARCO ANTONIO L. VELÁSQUEZ [Opens in a new window]
HENRIQUE F. DE LIMA*
Affiliation:
Departamento de MatemÁtica, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil e-mail: wfgomes.uepb@gmail.com, marco.velasquez@mat.ufcg.edu.br
WALLACE F. GOMES
Affiliation:
Departamento de MatemÁtica, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil e-mail: wfgomes.uepb@gmail.com, marco.velasquez@mat.ufcg.edu.br
MÁRCIO S. SANTOS
Affiliation:
Departamento de MatemÁtica, Universidade Federal da Paraíba, 58.051-900 João Pessoa, Paraíba, Brazil e-mail: marcio@mat.ufpb.br
MARCO ANTONIO L. VELÁSQUEZ
Affiliation:
Departamento de MatemÁtica, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil e-mail: wfgomes.uepb@gmail.com, marco.velasquez@mat.ufcg.edu.br
*
e-mail: henriquedelima74@gmail.com
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Abstract

We investigate geometric aspects of complete spacelike mean curvature flow solitons of codimension one in a generalized Robertson–Walker (GRW) spacetime $-I\times _{f}M^n$, with base $I\subset \mathbb R$, Riemannian fiber $M^n$ and warping function $f\in C^\infty (I)$. For this, we apply suitable maximum principles to guarantee that such a mean curvature flow soliton is a slice of the ambient space and to obtain nonexistence results concerning these solitons. In particular, we deal with entire graphs constructed over the Riemannian fiber $M^n$, which are spacelike mean curvature flow solitons, and we also explore the geometry of a conformal vector field to establish topological and further rigidity results for compact (without boundary) mean curvature flow solitons in a GRW spacetime. Moreover, we study the stability of spacelike mean curvature flow solitons with respect to an appropriate stability operator. Standard examples of spacelike mean curvature flow solitons in GRW spacetimes are exhibited, and applications related to these examples are given.

Keywords

GRW spacetimesspacelike mean curvature flow solitonsspacelike mean curvature flow soliton equationstable spacelike mean curvature flow solitons

MSC classification

Primary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 53C50: Lorentz manifolds, manifolds with indefinite metrics
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 116 , Issue 2 , April 2024 , pp. 221 - 256
Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first, third and fourth authors are partially supported by CNPq, Brazil, grants 301970/2019-0, 306524/2022-8 and 304891/2021-5, respectively. The second author is partially supported by CAPES, Brazil. The third author is also partially supported by FAPESQ-PB, Brazil, grant 3025/2021.

Communicated by J. McCoy

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