TY - JOUR
T1 - Noncompact shrinking four solitons with nonnegative curvature
AU - Naber, Aaron
PY - 2010/8/1
Y1 - 2010/8/1
N2 - 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.
AB - 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.
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U2 - 10.1515/CRELLE.2010.062
DO - 10.1515/CRELLE.2010.062
M3 - Article
AN - SCOPUS:78149264367
SN - 0075-4102
SP - 125
EP - 153
JO - Journal fur die Reine und Angewandte Mathematik
JF - Journal fur die Reine und Angewandte Mathematik
IS - 645
ER -