Noncompact shrinking four solitons with nonnegative curvature

作者：Naber Aaron^*

来源：Journal fur die Reine und Angewandte Mathematik, 2010, 645(645): 125-153.

DOI：10.1515/CRELLE.2010.062

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 R(4) or a finite quotient of S(2) x R(2) or S(3) x R. In the process we also show that a complete shrinking soliton (M, g, X) with bounded curvature is gradient and kappa-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.

• 出版日期2010-8