Capillary minimal slicing and scalar curvature rigidity

Abstract

We develop minimal slicing via capillary hypersurfaces to understand scalar curvature metric on manifolds bounded below with boundary. 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. Moreover, we prove a comparison and rigidity result of manifolds with Listing type scaling invariant scalar and mean curvature comparison : $Rg\ge -12g_{{\mathbb{H}}^4}$ and ${\hat{H}}^{2}g\ge {\hat{H}}_0{g}_{{\mathbb{H}}^4}$, where $\hat{H}$ is a shifted mean curvature depending on the scale of manifold. This proves the traced version of Gromov's conjecture on comparison and rigidity of hyperbolic convex domains

arXiv comment41 pages, included generalizations to negative curvature metrics, comments are welcome!