Classification of expanding and steady Ricci solitons with integral curvature decay

Catino, Giovanni¹ ; Mastrolia, Paolo² ; Monticelli, Dario³

¹ Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
² Dipartimento di Matematica, Università Degli Studi di Milano, Via Saldini 50, %deleted Cesare I-20133 Milano, %paper had Milano, Italy 20133 Italy
³ Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, %paper had Milano, Italy 20133 Italy

Geometry & topology, Tome 20 (2016) no. 5, pp. 2665-2685.

Résumé

In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian manifold. In particular we show that the only complete expanding solitons with nonnegative sectional curvature and integrable scalar curvature are quotients of the Gaussian soliton, while in the steady case we prove rigidity results under sharp integral scalar curvature decay. As a corollary, we obtain that the only three-dimensional steady solitons with less than quadratic volume growth are quotients of ${ℝ}^3$ or of $ℝ\times {\Sigma }^2$, where ${\Sigma }^2$ is Hamilton’s cigar.

Détail

DOI : 10.2140/gt.2016.20.2665

Classification : 53C20, 53C25
Keywords: Ricci solitons, Weitzenböck formula, weighted Einstein tensor, rigidity results

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