On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes

Giovanni Molica Bisci; José H. H. de Lacerda; Henrique F. de Lima; Marco A. L. Velásquez

doi:10.3934/dcdss.2023045

Article Contents

2023, Volume 16, Issue 11: 3212-3257. Doi: 10.3934/dcdss.2023045

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On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes

1.
Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino Carlo Bo, Urbino, 61029, Italy
2.
Departamento de Matemática, Universidade Federal de Campina Grande, 58.429-970 Campina Grande, Paraíba, Brazil

^*Corresponding author: Giovanni Molica Bisci
^*Corresponding author: Giovanni Molica Bisci

Dedicated to Vicenţiu D. Rădulescu with great esteem and admiration on the occasion of his 65th birthday

Received: September 2022

Revised: January 2023

Early access: March 2023

Published: November 2023

The first author is partially supported by INdAM-GNAMPA Research Project 2020 titled Equazioni alle derivate parziali: problemi e modelli. The second author is partially supported by CAPES, Brazil. The third and fourth authors are partially supported by CNPq, Brazil, grants 301970/2019-0 and 304891/2021-5, respectively

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Abstract

We study some geometric aspects of the higher order mean curvatures (or, more simply, the so-called $ r $-th mean curvatures) of a spacelike hypersurface immersed in a pp-wave spacetime, namely, in a connected Lorentzian manifold admitting a parallel and lightlike vector field. Initially, we develop general Minkowski-type integral formulas for compact (without boundary) spacelike hypersurfaces and we apply them to the study of the uniqueness and nonexistence of compact spacelike hypersurfaces in terms of their $ r $-mean curvatures. Next, we obtain a characterization of $ r $-stability for $ r $-maximal compact spacelike hypersurfaces through of the analysis of the first nonzero eigenvalue of an differential operator naturally attached to the $ r $-th mean curvature. For the noncompact case, by applying new forms of maximum principles on complete noncompact Riemannian manifolds due to Caminha [17] and Alías, Caminha and Nascimento [3], we obtain sufficient geometric conditions involving some $ r $-th mean curvature and the volume growth that allow us to establish some nonexistence results or to guarantee that a complete noncompact spacelike hypersurface is either totally geodesic, or totally umbilical, or maximal, or $ r $-maximal. We also obtain estimates for the index of minimum relative nullity of spacelike hypersurfaces.

Keywords:

pp-wave spacetimes,
spacelike hypersurfaces,
r-th mean curvatures,
Minkowski-type integral formulas,
r-stability,
Calabi-Bernstein-type results,
exponencial and polynomial volume growth.

Mathematics Subject Classification: Primary: 53C42, 53C44; Secondary: 53C50, 53Z05.

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