Volume 14, issue 4 (2010)

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On the classification of gradient Ricci solitons

Peter Petersen and William Wylie

Geometry & Topology 14 (2010) 2277–2300
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 ${S}^{n}$, ${S}^{n-1}×ℝ$ 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 ${ℍ}^{n}$, ${ℍ}^{n-1}×ℝ$, ${ℝ}^{n}$, ${S}^{n-1}×ℝ$ or ${S}^{n}$.

Keywords
Ricci soliton, Weyl tensor, locally conformally flat, three manifold, constant scalar curvature
Primary: 53C25
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