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Extremal and weakly trapped submanifolds in Generalized Robertson-Walker spacetimes

José A. S. Pelegrín

Abstract

In this article we obtain new rigidity results for spacelike submanifolds of arbitrary codimension in Generalized Robertson-Walker spacetimes. Namely, under appropriate assumptions such as parabolicity we prove by means of some maximum principles that they must be contained in a spacelike slice. This enables us to characterize extremal and weakly trapped submanifolds in these ambient spacetimes.

Extremal and weakly trapped submanifolds in Generalized Robertson-Walker spacetimes

Abstract

In this article we obtain new rigidity results for spacelike submanifolds of arbitrary codimension in Generalized Robertson-Walker spacetimes. Namely, under appropriate assumptions such as parabolicity we prove by means of some maximum principles that they must be contained in a spacelike slice. This enables us to characterize extremal and weakly trapped submanifolds in these ambient spacetimes.
Paper Structure (7 sections, 13 theorems, 27 equations)

This paper contains 7 sections, 13 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Parabolic Riemannian manifolds
  4. Submanifolds contained in spacelike slices
  5. Main results
  6. Rigidity results for monotonic warping function
  7. Rigidity results for bounded mean curvature

Key Result

Proposition 1

Let $\psi: M \longrightarrow \overline{M} = I \times_f F$ be a submanifold contained in a spacelike slice $\{t_0\} \times F$, $t_0 \in I$ of a GRW spacetime with causal mean curvature vector field $\overrightarrow{H}$. Then, $\psi(M)$ is

Theorems & Definitions (16)

  • Proposition 1
  • Proposition 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Theorem 6
  • Corollary 7
  • Corollary 8
  • Corollary 9
  • Remark 10
  • ...and 6 more