Geometry & Topology Volume 14, issue 4 (2010)

We show that the only shrinking gradient solitons with vanishing Weyl tensor and Ricci tensor satisfying a weak integral condition are quotients of the standard ones Sⁿ, Sⁿ⁻¹ × R and Rⁿ. This gives a new proof of the Hamilton–Ivey–Perelman classification of 3–dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of Hⁿ, Hⁿ⁻¹ × R, Rⁿ, Sⁿ⁻¹ × R or Sⁿ.

Keywords

Ricci soliton, Weyl tensor, locally conformally flat, three manifold, constant scalar curvature

Mathematical Subject Classification

Primary: 53C25

References

Publication

Received: 26 June 2008
Accepted: 30 August 2010
Published: 29 October 2010
Proposed: Walter Neumann
Seconded: Tobias Colding, Steven Ferry

Authors

Peter Petersen
Department of Mathematics University of California, Los Angeles 520 Portola Plaza Los Angeles CA 90095 USA
http://www.math.ucla.edu/~petersen
William Wylie
Department of Mathematics University of Pennsylvania David Rittenhouse Lab 209 South 33rd Street Philadelphia PA 19104 USA
http://www.math.upenn.edu/~wylie