**2/29/2008**

**Aaron Naber
Princeton University**

**The Structure of Shrinking Solitons**

We discuss a handful of structure theorems for shrinking solitons with bounded curvature. In particular we prove a priori injectivity radi.

We discuss a handful of structure theorems for shrinking solitons, in particular let (M,g,X) be a complete shrinking soliton with bounded curvature, then there exists k>0 and a smooth function f such that (M,g,X) is k-noncollapsed and (M,g,f) is a gradient shrinking soliton, generalizing results from the compact case. If M is a noncompact four dimensional shrinking soliton with nonnegative curvature then up to finite quotient it is isometric to R^4,RxS^3 or S^2xR^2. Finally we show that the singularity dilation of a Type I singularity on a Ricci Flow is a shrinking soliton.