Steady gradient Ricci soliton with curvature in $L^1$

We characterize complete nonnegatively curved steady gradient soliton with curvature in $L^1$.We show that there are isometric to a product $((\mathbb{R}^2,g_{{\rm cigar}})\times(\mathbb{R}^{n-2}, \eucl))/\Gamma$ where $\Gamma$ is a Bieberbach group of rank $n-2$.We prove also a similar local splitting result under weaker curvature assumptions.

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Published 21 March 2012