A splitting theorem for 3-manifold with nonnegative scalar curvature and mean-convex boundary

TL;DR

This paper proves a splitting theorem for 3-manifolds with nonnegative scalar curvature and mean-convex boundary, showing flatness under the presence of certain minimal surfaces, extending classical results in geometric analysis.

Contribution

It introduces new splitting results for 3-manifolds with boundary based on the existence of absolutely area-minimizing surfaces, including half-cylinders and strips, under various energy conditions.

Findings

Manifolds with specified minimal surfaces are flat.

Results extend classical splitting theorems to manifolds with boundary.

Applicable under energy-minimizing conditions for certain surfaces.

Abstract

We show that a Riemannian 3-manifold with nonnegative scalar curvature and mean-convex boundary is flat if it contains an absolutely area-minimizing (in the free boundary sense) half-cylinder or strip. Analogous results also hold for a $\theta$-energy-minimizing half-cylinder, or, under certain topological assumptions, a $\theta$-energy-minimizing strip for $\theta\in (0,\pi)$.