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

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.

On the umbilicity of linear Weingarten spacelike submanifolds immersed in the de Sitter space

We investigate the umbilicity of [Formula: see text]-dimensional complete linear Weingarten spacelike submanifolds immersed with parallel normalized mean curvature vector field in the de Sitter space [Formula: see text] of index [Formula: see text]. We recall that a spacelike submanifold is said to be linear Weingarten when its mean curvature function [Formula: see text] and its normalized scalar curvature [Formula: see text] satisfy a linear relation of the type [Formula: see text], for some constants [Formula: see text]. Under suitable constraints on the values of [Formula: see text] and [Formula: see text], we apply a generalized maximum principle for a modified Cheng–Yau operator [Formula: see text] in order to show that such a spacelike submanifold must be either totally umbilical or isometric to a product [Formula: see text], where the factors [Formula: see text] are totally umbilical submanifolds of [Formula: see text] which are mutually perpendicular along their intersections. Moreover, we also study the case in which these spacelike submanifolds are [Formula: see text]-parabolic.

An Isoperimetric-Type Inequality for Spacelike Submanifold in the Minkowski Space

We prove an isoperimetric-type inequality for maximal, spacelike submanifold in the Minkowski space. The argument is based on the recent work of Brendle.

Complete linear Weingarten spacelike submanifolds with higher codimension in the de Sitter space

We study complete linear Weingarten spacelike submanifolds with arbitrary high codimension [Formula: see text] in the de Sitter space [Formula: see text] of index [Formula: see text] and whose normalized mean curvature vector is parallel. Under suitable restrictions on the values of the mean curvature function and on the norm of the traceless part of the second fundamental form, we prove that such a spacelike submanifold must be either totally umbilical or isometric to a certain hyperbolic cylinder of [Formula: see text]. Our approach is based on the use of a Simons type formula related to an appropriate Cheng–Yau modified operator jointly with an extension of Hopf’s maximum principle for complete Riemannian manifolds.

Willmore spacelike submanifolds in an indefinite space form Nn+p q(c)

Let Nn+p q(c) be an (n+p)-dimensional connected indefinite space form of index q(1 ? q ? p) and of constant curvature c. Denote by ? : M ? Nn+p q (c) the n-dimensional spacelike submanifold in Nn+p q (c), ? : M ? Nn+p q(c) is called a Willmore spacelike submanifold in Nn+p q(c) if it is a critical submanifold to the Willmore functional W(?) = ?q M ?n dv =?M (S-nH2)n/2 dv, where S and H denote the norm square of the second fundamental form and the mean curvature of M and ?2 = S ? nH2. If q = p, in [14], we proved some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form Nn+p q(c). In this paper, we continue to study this topic and prove some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form Nn+p q(c) (1 ? q ? p).