Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T14:23:04.722Z Has data issue: false hasContentIssue false
  • English
  • Français

On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in Lorentz–Minkowski spacetime 𝕃m

Part of: Global differential geometry Partial differential equations on manifolds; differential operators Spectral theory and eigenvalue problems

Published online by Cambridge University Press:  11 February 2021

Francisco J. Palomo Open the ORCID record for Francisco J. Palomo [Opens in a new window]  and
Alfonso Romero Open the ORCID record for Alfonso Romero [Opens in a new window]
Francisco J. Palomo
Affiliation:
Departamento de MatemĂĄtica Aplicada, Universidad de MĂĄlaga, 29071MĂĄlaga, Spain (fjpalomo@ctima.uma.es)
Alfonso Romero
Affiliation:
Departamento de GeometrĂ­a y TopologĂ­a, Universidad de Granada, 18071Granada, Spain (aromero@ugr.es)
Get access Rights & Permissions [Opens in a new window]

Abstract

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension â©Ÿ2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

Keywords

Compact spacelike submanifoldsfirst eigenvalue of the Laplace operatormean curvature vector field

MSC classification

Primary: 53C40: Global submanifolds 35P15: Estimation of eigenvalues, upper and lower bounds 53C50: Lorentz manifolds, manifolds with indefinite metrics
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) 58J50: Spectral problems; spectral geometry; scattering theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 311 - 330
Check for updates
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alías, L. J., Estudillo, F. J. M. and Romero, A.. Spacelike submanifolds with parallel mean curvature in pseudo-Riemannian space forms. Tsukuba J. Math. 21 (1997), 169–179.CrossRefGoogle Scholar
Berger, M., Gauduchon, P. and Mazet, E.. Le spectre d'une variété Riemannienne. Lectures Notes in Mathematics, vol. 194 (Berlin-New York: Springer-Verlag, 1971).CrossRefGoogle Scholar
Bleecker, D. D. and Weiner, J. L.. Extrinsic bounds on λ1 of Δ on a compact manifold. Comment. Mat. Helvetici 51 (1976), 601–609.CrossRefGoogle Scholar
Chavel, I.. Eigenvalues in Riemannian Geometry. Pure and Applied Math, vol. 115 (Orlando, FL: Academic Press, 1984).Google Scholar
Chavel, I.. Riemannian Geometry. A modern Introduction, Second Edition. Cambridge Studies in Advances Mathematics, vol. 98 (Cambridge: Cambridge University Press, 2006).CrossRefGoogle Scholar
El Soufi, A. and Ilias, S.. Une inĂ©galitĂ© du type ‘Reilly’ pour les sousvariĂ©tĂ©s de l'espace hyperbolique. Comment. Mat. Helvetici 67 (1992), 167–181.CrossRefGoogle Scholar
GutiĂ©rrez, M., Palomo, F. J. and Romero, A.. A Berger–Green type inequality for compact Lorentzian manifolds. Trans. Am. Math. Soc. 354 (2002), 4505–4523.CrossRefGoogle Scholar
Harris, S. G.. A characterization of Robertson-Walker spaces by null sectional curvature. Gen. Relativ. Gravitation 17 (1985), 493–498.CrossRefGoogle Scholar
Heintze, E.. Extrinsic upper bound for λ1. Math. Ann. 280 (1988), 389–402.CrossRefGoogle Scholar
Hersch, J.. Quatre propriĂ©tĂ©s isoperimĂ©triques de membranes sphĂ©riques homogeĂšs. C. R. Acad. Sci. Paris Sr. A 270 (1970), 1645–1648.Google Scholar
Ishihara, T.. Maximal spacelike submanifolds of a pseudo-Riemannian space of constant curvature. Michigan Math. J. 35 (1988), 345–352.CrossRefGoogle Scholar
Kossowski, M.. The total split curvatures of knotted space-like 2-spheres in Minkowski 4-space. Proc. Am. Math. Soc. 117 (1993), 813–818.Google Scholar
Markvorsen, S.. A characteristic eigenfunction for minimal hypersurfaces in spaceforms. Math. Z. 202 (1989), 375–382.CrossRefGoogle Scholar
Palmas, O., Palomo, F. J. and Romero, A.. On the total mean curvature of a compact space-like submanifold in Lorentz–Minkowski spacetime. P. R. Soc. Edinb. A Mat. 148A (2018), 199–210.CrossRefGoogle Scholar
Palomo, F. J. and Romero, A.. On spacelike surfaces in 4-dimensional Lorentz–Minkowski spacetime through a lightcone. P. R. Soc. Edinb. A Mat. 143A (2013), 881–892.CrossRefGoogle Scholar
Reilly, R. C.. On the first eigenvalue of the Laplace operator for compact submanifolds of Euclidean space. Comment. Mat. Helvetici 52 (1977), 525–533.CrossRefGoogle Scholar
Takahashi, T.. Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18 (1966), 380–385.CrossRefGoogle Scholar
Treibergs, A. E.. Entire spacelike hypersurfaces of constant mean curvature in Minkowski space. Invent. Math. 66 (1982), 39–56.CrossRefGoogle Scholar