The dihedral rigidity conjecture for $n$-prisms

Abstract

We prove the following comparison theorem for metrics with nonnegative scalar curvature, also known as the dihedral rigidity conjecture by Gromov: for $n \leq 7$, if an $n$-dimensional prism has nonnegative scalar curvature and weakly mean convex faces, then its dihedral angle cannot be everywhere not larger than its Euclidean model, unless it is isometric to an Euclidean prism. The proof relies on constructing certain free boundary minimal hypersurface in a Riemannian polyhedron, and extending a dimension descent idea of Schoen–Yau. Our result is a localization of the positive mass theorem.