Bonnet-Myers theorem: Given a smooth Riemannian manifold, if the Ricci curvature is greater than a uniform positive constant at every point, then the manifold is compact.


There are many metric surfaces that are not smooth. The most basic example is the tetrahedron surface, which is not smooth at the vertices. More generally, polygonal surfaces are surfaces obtained by gluing regular polygons along edges. In these talks, we show that if the "singular curvature" at each vertex of the polygonal surface is greater than a uniform positive constant, then the polygonal surface is compact! The key idea is to "distribute" the positive curvature accumulated at each vertex over the entire surface.