# The Structure of Shrinking Solitons

# 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 $k\gt 0$ 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 $R^4$,$R\times S^3$ or $S^2\times R^2$. Finally we show that the singularity dilation of a Type I singularity on a Ricci Flow is a shrinking soliton.