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 $\theta \in (0,\pi)$.
This work proves scalar-curvature splitting theorems for noncompact three-manifolds with mean-convex boundary, using noncompact minimizing free-boundary and capillary surfaces as the rigidifying objects.
Free-boundary splitting
If the manifold contains an absolutely area-minimizing free-boundary half-cylinder or strip, then it is flat. Up to scaling, the resulting models are covered by $\mathbb S^1\times\mathbb R^2_+$ or are isometric to $[0,1]\times\mathbb R^2$.
Capillary extension
For $\theta\in(0,\pi)$, an absolutely $\theta$-energy-minimizing half-cylinder gives the corresponding tilted flat model; the strip statement also holds when its two boundary curves lie on different boundary components.
No compact-boundary hypothesis
The argument requires neither compactness nor a prescribed number of components of $\partial M$, extending earlier boundary splitting theorems beyond their usual global assumptions.
Plateau-type problems in carefully chosen prism-shaped domains produce a sequence of minimizing capillary surfaces near the given noncompact surface. Stability classifies the possible limits, while energy comparisons rule out unwanted disk, half-plane, plane, and cylinder degenerations. The surviving half-cylinders or strips form a rigid family; equality in the scalar-curvature and boundary inequalities then propagates to an ambient flat product.
My research interests include Geometric Analysis and Partial Differential Equations.