On a dichotomy of the curvature decay of steady Ricci solitons

Abstract

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci solitons with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher dimensions under the additional assumption that the Ricci curvature is nonnegative outside a compact subset. As an application, we prove some classification results on steady gradient Ricci solitons with fast curvature decay and obtain a dichotomy on the asymptotic geometry at spatial infinity.