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

## TL;DR

This paper shows that classical eigenvalue bounds do not apply to spacelike submanifolds in Lorentz-Minkowski spacetime, introduces new bounds, and characterizes when equality holds in geometric terms.

## Contribution

It develops a new integral technique for Lorentzian geometry and establishes novel extrinsic upper bounds for the first eigenvalue of spacelike submanifolds.

## Key findings

- Reilly's bound does not hold in Lorentz-Minkowski spacetime.
- New upper bounds for the first eigenvalue are established.
- Equality characterizes submanifolds minimally embedded in hyperspheres.

## 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.