Classification of expanding and steady Ricci solitons with integral curvature decay

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 $\mathbb{R}\times {\Sigma }^2$, where ${\Sigma }^2$ is Hamilton's cigar.

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• O(2)-symmetry of 3D steady gradient Ricci solitons
• Curvature estimates for steady Ricci solitons
• On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization
• Classification of 3-dimensional complete rectifiable steady and expanding gradient Ricci solitons
• On the construction of complete expanding gradient Rici solitons