On a dichotomy of the curvature decay of steady Ricci soliton

Abstract

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton 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.

Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons

“…Remark 5.2. Moreover, based on the constructions of Deruelle [32,33], very recently Chan and Zhu [19] have exhibited an example of 3-dimensional complete noncompact asymptotically conical gradient expanding soliton with nonnegative curvature operator Rm ≥ 0 such that lim inf Moreover, in the shrinking case, the estimate |∇Rm| ≤ CR of Munteanu and Wang [46] implies the sharper gradient estimate |∇ ln R| ≤ C on M.…”