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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Classification of expanding and steady Ricci solitons with integral curvature decay

Giovanni Catino, Paolo Mastrolia and Dario D Monticelli

Geometry & Topology 20 (2016) 2665–2685

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 3 or of × Σ2, where Σ2 is Hamilton’s cigar.

Ricci solitons, Weitzenböck formula, weighted Einstein tensor, rigidity results
Mathematical Subject Classification 2010
Primary: 53C20, 53C25
Received: 7 January 2015
Revised: 23 September 2015
Accepted: 9 November 2015
Published: 7 October 2016
Proposed: Gang Tian
Seconded: John Lott, Tobias H. Colding
Giovanni Catino
Dipartimento di Matematica
Politecnico di Milano
Piazza Leonardo da Vinci 32
I-20133 Milano
Paolo Mastrolia
Dipartimento di Matematica
Università Degli Studi di Milano
Via Saldini 50
%deleted Cesare I-20133 Milano
%paper had Milano, Italy 20133 Italy
Dario D Monticelli
Dipartimento di Matematica
Politecnico di Milano
Piazza Leonardo da Vinci 32
I-20133 Milano
%paper had Milano, Italy 20133 Italy