Rigidity of asymptotically conical shrinking gradient Ricci solitons

Abstract

We show that if two gradient shrinking Ricci solitons are asymptotic along some end of each to the same regular cone $((0,\infty) \times \Sigma, dr^2 + r^2 g_{\Sigma} )$, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on the behavior of the metrics off of the ends in question and in particular does not require their geodesic completeness. As an application, we prove that the only complete connected gradient shrinking Ricci soliton asymptotic to a rotationally symmetric cone is the Gaussian soliton on $\mathbb{R}^n$.