Noncompact shrinking four solitons with nonnegative curvature

Aaron Naber*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

104 Scopus citations

Abstract

We prove that if (M, g, X) is a noncompact four dimensional shrinking soliton with bounded nonnegative curvature operator, then (M, g) is isometric to or a finite quotient of or S3 × . In the process we also show that a complete shrinking soliton (M, g, X) with bounded curvature is gradient and κ-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. The proofs rely on the technical construction of a singular reduced length function, a function which behaves as the reduced length function but can be extended to singular times.

Original languageEnglish (US)
Pages (from-to)125-153
Number of pages29
JournalJournal fur die Reine und Angewandte Mathematik
Issue number645
DOIs
StatePublished - Aug 1 2010

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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