Rigidity results for free boundary hypersurfaces in initial data sets with boundary
Deivid de Almeida, Abraão Mendes
TL;DR
The paper develops sharp rigidity results for free boundary hypersurfaces in initial data sets with boundary under lower scalar curvature and energy conditions. By extending warped-product splitting techniques to manifolds with boundary and employing free boundary MOTS stability together with Yamabe invariant methods, it derives product-type local geometries and volume/ridigity bounds. The core contributions include a new warped-product extension (Theorem A) and a volume rigidity theorem for free boundary MOTS with negative Yamabe invariant (Theorem B), both featuring precise equality cases that force Einstein or Ricci-flat cross-sections and totally geodesic boundaries. The results deepen the understanding of scalar curvature rigidity in the presence of boundaries and have potential implications for general relativity, particularly in the study of initial data sets with boundary. The approach integrates MOTS theory, parabolic maximum principles, and conformal geometry of manifolds with boundary to obtain these rigidity phenomena.
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.