Ricci solitons and collapsed spaces

Abstract

This thesis has two primary parts. In the first part we study shrinking Ricci solitons. We classify shrinking solitons in dimension four with bounded nonnegative curvature and in dimension three with bounded nonnegative Ricci curvature, thus extending Perelman's result from dimension three. We also prove structure theorems for shrinking solitons in any dimension. In particular we prove a shrinking soliton with bounded curvature is gradient and is a priori noncollapsed.

In the second part we study collapsing limits of Riemannian manifolds. We give sharp estimates on the Riemannian orbifold points of collapsed spaces X which are the limits of manifolds M_i with either sectional curvature bounds in any dimension or Ricci and topology bounds in dimension four. Then we build a structure, which we call an N*-bundle, over the collapsed space which allows for stronger notions of convergence M_i → X as well as a background tool for doing analysis on X. As an application we generalize Gromov's Almost Flat Theorem and prove new Ricci pinching theorems.