UC Berkeley

Perelman stated without proof that a 3-dimensional compact Riemannian manifold which is locally volume collapsed, with respect to a lower curvature bound, is a graph manifold. The theorem was used to complete his Ricci flow proof of Thurston’s geometrization conjecture. Kleiner and Lott gave a proof of the theorem as a part of their presentation of Perelman’s proof.

In this dissertation, we generalize Kleiner and Lott’s version of Perelman’s theorem to 4-dimensional closed Riemannian manifolds. We show that under some regularity assumptions, if a 4-dimensional closed Riemannian manifold is locally volume collapsed then it admits an F-structure or a metric of nonnegative sectional curvature.