Nonpositively Curved Manifolds of Small Volume

This thesis contributes a novel theorem in the mathematical literature of collapsing manifolds. We prove that for compact manifolds of nonpositive sectional curvature bounded below, with uniformly bounded diameter, and containing no local Euclidean factors, there is a uniform lower bound on volume. This lower volume bound only depends on the dimension of the manifolds and the uniform bound on diameter, and represents an obstruction to collapsing. The theorem is analogous to previous results in collapsing manifolds, in particular, a theorem of Kazhdan and Margulis on the minimal volume of locally symmetric spaces.