A splitting theorem for scalar curvature

TL;DR

This paper proves that a 3-dimensional Riemannian manifold with non-negative scalar curvature must be flat if it contains an area-minimizing cylinder, confirming a conjecture related to scalar curvature and manifold splitting.

Contribution

It establishes a scalar curvature analogue of the classical splitting theorem, proving flatness under the presence of an area-minimizing cylinder in 3-manifolds.

Findings

Manifolds with non-negative scalar curvature containing an area-minimizing cylinder are flat.

Confirms conjecture by Fischer-Colbrie, Schoen, Cai, and Galloway.

Extends classical splitting results to scalar curvature setting.

Abstract

We show that a Riemannian $3$-manifold with non-negative scalar curvature is flat if it contains an area-minimizing cylinder. This scalar-curvature analogue of the classical splitting theorem of J.~Cheeger and D.~Gromollhas been conjectured by D.~Fischer-Colbrie and R.~Schoen and by M.~Cai and G.~Galloway.