Hilbert Domains, Conics, and Rigidity

Abstract

A class of compact projective manifolds can be viewed as a convex set in projective space modulo a discrete group of isometries. This thesis explores the circumstances under which this convex set is a symmetric convex cone. The irreducible symmetric convex cones are analogous to symmetric spaces in Riemannian geometry and consist of hyperbolic space and positive definite Hermitian matrices. Having a properly embedded conic in the boundary of the convex set is equivalent to the existence of a subspace isometric to the hyperbolic plane. When enough of these conics exist, I will show that the convex set is a symmetric convex cone. This demonstrates how the shape of the boundary of the convex set determines its isometry class. Further, if enough twice differentiable curves are found in the boundary of the convex set, I will show that it must be hyperbolic space. This result also has applications to affine spheres.