Noncompact Shrinking 4-Solitons with Nonnegative Curvature

2007/10/30 by Aaron Naber, Naber, Aaron · 6 citations
Physics and Astronomy · Mathematics · Biochemistry, Genetics and Molecular Biology · #Advanced Differential Geometry Research #Geometric Analysis and Curvature Flows #Microbial metabolism and enzyme function

paper · pdf · doi:10.48550/arxiv.0710.5579

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 R⁴ or a finite quotient of S²xR² or S³xR. 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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