Volume 20, issue 5 (2016)

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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
Abstract

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 $ℝ×{\Sigma }^{2}$, where ${\Sigma }^{2}$ is Hamilton’s cigar.

Keywords
Ricci solitons, Weitzenböck formula, weighted Einstein tensor, rigidity results
Mathematical Subject Classification 2010
Primary: 53C20, 53C25