Capillary minimal slicing and scalar curvature rigidity

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 : $R g \ge -12 g_{\mathbb{H}^{4}}$ 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