Noncompact Shrinking 4-Solitons with Nonnegative Curvature

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.

30 pages

Cited by in corpus (10)

• Recent Progress on Ricci Solitons
• On Type I Singularities in Ricci flow
• Geometry of Complete Gradient Shrinking Ricci Solitons
• Remarks on curvature behavior at the first singular time of the Ricci flow
• Remarks on non-compact gradient Ricci solitons
• On gradient Ricci solitons with Symmetry
• On gradient Ricci solitons
• On the Perelman's reduced entropy and Ricci flat manifolds with maximal volume growth
• On the asymptotic reduced volume of the Ricci flow
• Convergence of compact Ricci solitons