On the classification of gradient Ricci solitons

Petersen, Peter¹ ; Wylie, William²

¹ Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Los Angeles CA 90095, USA
² Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia PA 19104, USA

Geometry & topology, Tome 14 (2010) no. 4, pp. 2277-2300.

Résumé

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 $S^n$.

Détail

DOI : 10.2140/gt.2010.14.2277

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

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