Compact maximal hypersurfaces in globally hyperbolic spacetimes

Abstract

Several uniqueness results on compact maximal hypersurfaces in a wide class of globally hyperbolic spacetimes are provided. They are obtained from the study of a distinguished function on the maximal hypersurface and under suitable natural first order conditions of the spacetime. As a consequence, several applications to Geometric Analysis are given.

Rigidity and non-existence of spacelike submanifolds with causal mean curvature vector field in spacetimes and the Cauchy problem in General Relativity

“…In this framework, we can describe a broad family of spacetimes (which includes, for instance, the hyperbolic case) by considering a differentiable manifold F , an open interval I ⊂ R and an one-parametric family {g t } t∈I of Riemannian metrics on F (see [2] for the details). Then, the product manifold M = I × F can be endowed with the Lorentzian metric given at each point (t, p) ∈ I × F by…”

On complete trapped submanifolds in globally hyperbolic spacetimes

“…Furthermore, the same authors proved in [10] that any Cauchy hypersurface of a globally hyperbolic spacetime M determines an orthogonal splitting as above. Motivated by the above splitting result, and following the notation in [3], we will say that a spacetime (M n+m+1 , g) is orthogonally splitted if it is isometric to a product manifold (1)…”