Fundamental Domains of Convex Projective Structures

Convex (or properly convex) projective structures on manifolds share many common features with non-positively curved metrics. The lack of invariant metrics, however, makes it harder to study them. For example, some of the well-known facts about fundamental domains in the case of constant curvature geometries are no longer obvious in projective geometry. In my talk, I will show that every properly convex projective structure admits a convex fundamental polyhedron, which is the Dirichlet domain with respect to a certain distance-like function. The proof makes an essential use of the solution (by Cheng and Yau) of Calabi’s conjecture on complete hyperbolic affine spheres and the duality relation between them.