Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A CMC existence result for expanding cosmological spacetimes

Gregory J. Galloway, Eric Ling

Abstract

We establish a new CMC (constant mean curvature) existence result for cosmological spacetimes, i.e., globally hyperbolic spacetimes with compact Cauchy surfaces satisfying the strong energy condition. If the spacetime contains an expanding Cauchy surface and is future timelike geodesically complete, then the spacetime contains a CMC Cauchy surface. This result settles, under certain circumstances, a conjecture of the authors and a conjecture of Dilts and Holst. Our proof relies on the construction of barriers in the support sense, and the CMC Cauchy surface is found as the asymptotic limit of mean curvature flow. Analogous results are also obtained in the case of a positive cosmological constant $Λ> 0$. Lastly, we include some comments concerning the future causal boundary for cosmological spacetimes which pertain to the CMC conjecture of the authors.

A CMC existence result for expanding cosmological spacetimes

Abstract

We establish a new CMC (constant mean curvature) existence result for cosmological spacetimes, i.e., globally hyperbolic spacetimes with compact Cauchy surfaces satisfying the strong energy condition. If the spacetime contains an expanding Cauchy surface and is future timelike geodesically complete, then the spacetime contains a CMC Cauchy surface. This result settles, under certain circumstances, a conjecture of the authors and a conjecture of Dilts and Holst. Our proof relies on the construction of barriers in the support sense, and the CMC Cauchy surface is found as the asymptotic limit of mean curvature flow. Analogous results are also obtained in the case of a positive cosmological constant . Lastly, we include some comments concerning the future causal boundary for cosmological spacetimes which pertain to the CMC conjecture of the authors.

Paper Structure

This paper contains 9 sections, 9 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Spacetime and mean curvature flow preliminaries
  3. Proof of Theorem \ref{['thm: main']}
  4. The case of a positive cosmological constant
  5. Extending Theorem \ref{['thm: main']} to the case $\Lambda > 0$
  6. Remarks on extending the results in Ecker_Huisken to the case $\Lambda > 0$
  7. Further comments on Conjecture \ref{['conj.']}
  8. Some remarks on the causal boundary of cosmological spacetimes
  9. Existence of a maximal Cauchy surface from a spacelike causal boundary

Key Result

Theorem 1

Let $(M,g)$ be a spacetime with compact Cauchy surfaces. Suppose $(M,g)$ is future timelike geodesically complete and has everywhere nonpositive timelike sectional curvatures, i.e., $K \leq 0$ everywhere. Then $(M,g)$ contains a CMC Cauchy surface.

Theorems & Definitions (11)

  • Theorem 1: GalLing
  • Conjecture 2
  • Theorem 3
  • Proposition 4
  • Proposition 5
  • Proposition 6
  • Theorem 7
  • Conjecture 8: Bartnik splitting conjecture
  • Proposition 9
  • Theorem 10
  • ...and 1 more