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 ⩾2) 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.

通过一个反例，我们证明了欧几里得空间紧子流形的拉普拉斯算子的第一个非平凡特征值的上限的 Reilly 定理（Reilly，1977，Comment . Mat. Helvetici，52，525 –533) 不适用于 Lorentz-Minkowski 时空的 (codimension ⩾2) 紧凑类空间子流形。在寻找替代结果时，应注意 (Reilly, 1977, Comment. Mat. Helvetici , 52 ) 中的原始技术, 525–533) 不适用于 Lorentz–Minkowski 时空的紧凑类空间子流形。在本文中，开发了一种基于 Lorentz-Minkowski 时空中光锥紧凑类空间截面积分公式的新技术。该技术在我们的环境中是真实的，即它不能扩展到另一个更高索引的半欧几里得空间。结果，得到了洛伦兹-闵可夫斯基时空紧致类空间子流形的拉普拉斯算子的第一特征值的上界族。这些不等式之一的等式具有几何特征。实际上，当且仅当紧致类空间子流形最小地位于某个类空间超平面的超球面中时，本征值才达到这些上界之一。在途中，