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

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

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 ([email protected])
Alfonso Romero
Affiliation:
Departamento de GeometrĂ­a y TopologĂ­a, Universidad de Granada, 18071Granada, Spain ([email protected])
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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
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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