Abstract: In 2013, Gromov proposed a dihedral rigidity conjecture, aiming at establishing a geometric comparison theory for metrics with positive scalar curvature. The conjecture states that if a Riemannian polyhedron has nonnegative scalar curvature in the interior, and weakly mean convex faces, then the dihedral angle between adjacent faces cannot be everywhere less than the corresponding Euclidean model. I will prove this conjecture for a large collection of polytopes. The strategy is to relate this conjecture with a geometric variational problem of capillary type, and apply the Schoen-Yau minimal slicing technique for manifolds with boundary.
