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On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in LorentzâMinkowski spacetime đm
Published online by Cambridge University Press:Â 11 February 2021
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, 525â533) does not work for a (codimension â©Ÿ2) compact spacelike submanifold of LorentzâMinkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525â533) is not applicable for a compact spacelike submanifold of LorentzâMinkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in LorentzâMinkowski 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 LorentzâMinkowski 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 LorentzâMinkowski spacetime through a spacelike hyperplane.
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- Research Article
- Information
- Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 311 - 330
- Copyright
- Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh