Capillary minimal slicing and scalar curvature rigidity

TL;DR

This paper introduces a new minimal slicing method using capillary hypersurfaces to analyze scalar curvature rigidity on manifolds with boundary, leading to a rigidity result in four dimensions under specific curvature conditions.

Contribution

It develops a novel approach combining capillary hypersurfaces and minimal slicing to establish scalar curvature rigidity results for manifolds with boundary, especially in four dimensions.

Findings

Proves a scalar curvature rigidity theorem for 4-manifolds with boundary.

Establishes conditions under which the manifold is isometric to a Euclidean domain.

Provides a new technique for understanding positive scalar curvature on manifolds with boundary.

Abstract

We develop minimal slicing via capillary hypersurfaces to understand positive scalar curvature metric on manifolds with boundary. The method provides rigidity statements once the regularity of minimizers of capillary area functional holds. In particular, in dimension $4$, we prove following comparison and rigidity statement: given a compact Riemannian $4$-manifold $(M^4,g)$ with a mean convex boundary whose boundary is diffeomorphic to boundary of a connected convex domain in $\mathbb R^4$, if the scalar curvature is non-negative and the scaled mean curvature comparison holds along the boundary, then $M$ is isometric to the Euclidean domain.