Rotational symmetry of non negatively curved expanding gradient Ricci solitons

Abstract

Let $(M^n,g,\nabla f)$, $n\ge 3$, be an expanding gradient Ricci soliton with nonnegative sectional curvature whose asymptotic cone is isometric to $C({\mathbb{S}}^{n-1}(c))$ where ${\mathbb{S}}^{n-1}(c)$ is the standard $(n-1)$-sphere of curvature $1/c^2$, with $c\in (0,1)$. We prove that if the convergence to the asymptotic cone is smooth, $(M^n,g,\nabla f)$ is rotationally symmetric. This is the expanding analogue of the Perelman conjecture on the Bryant soliton and this work is based on the proof by Brendle \cite{Bre-Rot-3d}. This has also been proved recently by Chodosh \cite{Cho-EGS}.

arXiv commentThis paper has been withdrawn by the author due to a an error in the analysis of the blow up of the Lichnerowicz equation