Initial Data Rigidity Results

Michael Eichmair,Gregory J Galloway,Abraão Mendes

doi:10.1007/s00220-021-04033-x

Michael Eichmair, Gregory J Galloway + Show 1 more

Open Access

https://doi.org/10.1007/s00220-021-04033-x

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Abstract

We prove several rigidity results related to the spacetime positive mass theorem. A key step is to show that certain marginally outer trapped surfaces are weakly outermost. As a special case, our results include a rigidity result for Riemannian manifolds with a lower bound on their scalar curvature.

Highlights

In this paper we establish several rigidity results for initial data sets that are motivated by the spacetime positive mass theorem
The DEC is the requirement that the scalar curvature of (M, g) be non-negative
Lohkamp has presented a different proof of Theorem 1.1 for all n ≥ 3. His method is by reduction to and proof of the following result: Let (M, g, K ) be an initial data set that is isometric to Euclidean space, with K = 0, outside some bounded open set U ⊂ M

Summary

Introduction

In this paper we establish several rigidity results for initial data sets that are motivated by the spacetime positive mass theorem. The initial data set is said to satisfy the dominant energy condition (DEC for short) if μ ≥ | J |. An initial data set (M, g, K ) is said to be time-symmetric or Riemannian if K = 0 In this case, the DEC is the requirement that the scalar curvature of (M, g) be non-negative. Lohkamp has presented a different proof of Theorem 1.1 for all n ≥ 3 His method is by reduction to and proof of the following result: Let (M, g, K ) be an initial data set that is isometric to Euclidean space, with K = 0, outside some bounded open set U ⊂ M. Note the similarity of Corollary 1.4 with the rigidity result Theorem 1 in [8] in the Ricci curvature setting, due to C.

Preliminaries

The Weakly Outermost Condition

Embedding of the Initial Data into a Quotient of Minkowski Space