Scalar curvature comparison and rigidity of $3$-dimensional weakly convex domains

For a compact Riemannian $3$-manifold $(M^3,g)$ with mean convex boundary which is diffeomorphic to a weakly convex compact domain in ${\mathbb{R}}^3$, we prove that if scalar curvature is nonnegative and the scaled mean curvature comparison $H^{2}g\ge H_0^2{g}_{Eucl}$ holds, then $(M,g)$ is flat. Our result is a smooth analog of Gromov’s dihedral rigidity conjecture and an effective version of extremity results on weakly convex balls in ${\mathbb{R}}^3$. More generally, we prove the comparison and rigidity theorem for several classes of manifold with corners. Our proof uses capillary minimal surfaces with prescribed contact angle together with the construction of foliation with nonnegative mean curvature and with prescribed contact angles.