Noncompact Shrinking 4-Solitons with Nonnegative Curvature

Details Noncompact Shrinking 4-Solitons with Nonnegative Curvature Noncompact Shrinking 4-Solitons with Nonnegative Curvature

2007-10-30

arxiv.org/abs/0710.5579v3

Mathematics

Differential Geometry

30 pages

Scientific paper

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.

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