# Some scalar curvature warped product splitting theorems

## TL;DR

This paper establishes new rigidity theorems for Riemannian manifolds with scalar curvature bounds and boundary conditions, using marginally outer trapped surfaces and Obata's equation, with implications for hyperbolic geometry.

## Contribution

It introduces novel scalar curvature splitting theorems for manifolds with boundary, extending rigidity results through techniques involving trapped surfaces and boundary analysis.

## Key findings

- Rigidity results for manifolds with scalar curvature ≥ -n(n-1) or ≥ 0.
- Use of marginally outer trapped surfaces in proving geometric rigidity.
- Analysis of Obata's equation on manifolds with boundary.

## Abstract

We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on marginally outer trapped surfaces applied to appropriate initial data sets. One of the results involves an analysis of Obata's equation on manifolds with boundary. This result is relevant to recent work of Lan-Hsuan Huang and the second author concerning the rigidity of asymptotically locally hyperbolic manifolds with zero mass.