On the First eigenvalue of the Laplace operator for Compact Spacelike submanifolds in Lorentz-Minkowski Spacetime $\mathbb{L}^{m}$

Abstract

By means of a family of counter-examples, it is shown that the Reilly upper bound for the first eigenvalue of the Laplace operator for a compact submanifold in Euclidean space does not work for $n$-dimensional compact spacelike submanifolds of Lorentz-Minkowski spacetime $\mathbb{L}^m$, $m\geq n+2$. We develop a new suitable technique, based on an integral formula on compact spacelike sections of the light cone in $\mathbb{L}^m$. Then, a family of extrinsic upper bounds for the first eigenvalue of the Laplace operator for a compact spacelike submanifold in $\mathbb{L}^m$ is proved. For each one of these inequalities, becoming an equality can be characterized in geometric terms. In particular, the eigenvalue achieves one of these upper bounds if and only if the submanifold lies minimally in certain hypersphere of a spacelike hyperplane.