Noncompact Shrinking 4-Solitons with Nonnegative Curvature

TL;DR

This paper classifies noncompact four-dimensional shrinking solitons with nonnegative curvature, showing they are isometric to standard models, and establishes properties of shrinking solitons with bounded curvature, including their gradient nature and collapse behavior.

Contribution

It provides a classification of 4D shrinking solitons with nonnegative curvature and proves that bounded curvature shrinking solitons are gradient and k-noncollapsed, also analyzing singularity dilations.

Findings

Classified 4D shrinking solitons with nonnegative curvature as R^4, S^2xR^2, or S^3xR.

Proved that bounded curvature shrinking solitons are gradient and k-noncollapsed.

Showed that Type I singularity dilations are shrinking solitons.

Abstract

We prove the following: Let (M,g,X) be a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M,g) is isometric to R^4 or a finite quotient of S^2xR^2 or S^3xR. In the process we also show that a complete shrinking soliton (M,g,X) with bounded curvature is gradient and k-noncollapsed and the dilation of a Type I singularity is a shrinking soliton. Further in dimension three we show shrinking solitons with bounded curvature can be classified under only the assumption of Rc>= 0.