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On complete trapped submanifolds in globally hyperbolic spacetimes

Alma L. Albujer, Jónatan Herrera, Rafael M. Rubio

Abstract

The aim of this manuscript is to obtain rigidity and non-existence results for parabolic spacelike submanifolds with causal mean curvature vector field in orthogonally splitted spacetimes, and in particular, in globally hyperbolic spacetimes. We also obtain results regarding the geometry of submanifolds by ensuring, under some mild hypothesis, the non-existence of local minima or maxima of certain distinguished function. Furthermore, in this last case the submanifold does not need to be parabolic or even complete. As an application in General Relativity, we obtain several nice results regarding (non-necessarily closed) trapped surfaces in a huge family of spacetimes. In fact, we show how our technique allows us to recover some relevant previous results for trapped surfaces in both, standard static spacetimes and Generalized Robertson-Walker spacetimes.

On complete trapped submanifolds in globally hyperbolic spacetimes

Abstract

The aim of this manuscript is to obtain rigidity and non-existence results for parabolic spacelike submanifolds with causal mean curvature vector field in orthogonally splitted spacetimes, and in particular, in globally hyperbolic spacetimes. We also obtain results regarding the geometry of submanifolds by ensuring, under some mild hypothesis, the non-existence of local minima or maxima of certain distinguished function. Furthermore, in this last case the submanifold does not need to be parabolic or even complete. As an application in General Relativity, we obtain several nice results regarding (non-necessarily closed) trapped surfaces in a huge family of spacetimes. In fact, we show how our technique allows us to recover some relevant previous results for trapped surfaces in both, standard static spacetimes and Generalized Robertson-Walker spacetimes.
Paper Structure (7 sections, 14 theorems, 49 equations)

This paper contains 7 sections, 14 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. On orthogonally splitted spacetimes
  3. On submanifolds in orthogonally splitted spacetimes
  4. Parabolicity of spacelike submanifolds
  5. On the geometry of spacelike submanifolds in an orthogonally splitted spacetime
  6. On trapped surfaces in orthogonally splitted spacetimes
  7. Application to cosmological models of interest

Key Result

Proposition 2.1

Let $(\overline{M}^{n+m+1},\overline{g})$ be an orthogonally splitted spacetime with $\beta\equiv 1$. Let $F_{t_0}$ be a spacelike slice whose shape operator is positive (resp. negative) semi-definite. Assume also that the sectional curvature of timelike planes of the spacetime are non-negative. The

Theorems & Definitions (30)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Definition 3.1
  • Proposition 4.1
  • proof
  • Remark 4.2
  • Corollary 4.3
  • Theorem 5.1
  • proof
  • ...and 20 more