On the classification of gradient Ricci solitons

Petersen, Peter; Wylie, William

doi:10.2140/gt.2010.14.2277

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On the classification of gradient Ricci solitons

Peter Petersen and William Wylie

Geometry & Topology 14 (2010) 2277–2300

DOI: 10.2140/gt.2010.14.2277

Abstract

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 $S^{n}$ , $S^{n - 1} \times ℝ$ and $ℝ^{n}$ . 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 $ℍ^{n}$ , $ℍ^{n - 1} \times ℝ$ , $ℝ^{n}$ , $S^{n - 1} \times ℝ$ or $S^{n}$ .

Keywords

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

Mathematical Subject Classification 2000

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

Volume 14, issue 4 (2010)