Rigidity results for free boundary hypersurfaces in initial data sets with boundary

TL;DR

This paper establishes new rigidity theorems for free boundary hypersurfaces in initial data sets with boundary, extending previous splitting and scalar curvature results to manifolds with boundary using free boundary MOTS techniques.

Contribution

It extends local splitting theorems and scalar curvature rigidity results to initial data sets with boundary, incorporating free boundary MOTS methods.

Findings

Extended local splitting theorems to manifolds with boundary.

Derived rigidity results for free boundary MOTS in initial data sets.

Connected scalar curvature conditions with free boundary hypersurface geometry.

Abstract

In this work, we present several rigidity results for compact free boundary hypersurfaces in initial data sets with boundary. Specifically, in the first part of the paper, we extend the local splitting theorems from [G. J. Galloway and H. C. Jang, Some scalar curvature warped product splitting theorems, Proc. Am. Math. Soc. 148 (2020), no. 6, 2617-2629] to the setting of manifolds with boundary. To achieve this, we build on the approach of the original paper, utilizing results on free boundary marginally outer trapped surfaces (MOTS) applied to specific initial data sets. In the second part, we extend the main results from [A. Barros and C. Cruz, Free boundary hypersurfaces with non-positive Yamabe invariant in mean convex manifolds, J. Geom. Anal. 30 (2020), no. 4, 3542-3562] to the context of free boundary MOTS in initial data sets with boundary.