On Area Comparison and Rigidity Involving the Scalar Curvature

TL;DR

This paper establishes a splitting theorem for Riemannian manifolds with scalar curvature bounds and area-minimising hypersurfaces, generalizing previous results and exploring their optimality through explicit examples.

Contribution

It introduces a new splitting theorem for manifolds with scalar curvature bounds involving area-minimising hypersurfaces, extending prior work by Nunes.

Findings

Proves a splitting theorem for manifolds with scalar curvature bounded below by a negative constant.

Develops an area comparison theorem for hypersurfaces with non-positive Sigma-constant.

Constructs examples demonstrating the optimality of the comparison and splitting results.

Abstract

We prove a splitting theorem for Riemannian n-manifolds with scalar curvature bounded below by a negative constant and containing certain area-minimising hypersurfaces (Theorem 3). Thus we generalise [25,Theorem 3] by Nunes. This splitting result follows from an area comparison theorem for hypersurfaces with non-positive Sigma-constant (Theorem 4) that generalises [23, Theorem 2]. Finally, we will address the optimality of these comparison and splitting results by explicitly constructing several examples.