Hamilton-Ivey estimates for gradient Ricci solitons

Abstract

We first show that any 4-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy |Rm| ≤ cR for some positive constant c. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound of the curvature operator for 4-dimensional steady gradient solitons with linear scalar curvatrue decay and proper potential function. The technique is also used to establish a sufficient condition for a 3-dimensional expanding gradient Ricci soliton to have positive curvature. This sufficient condition is satisfied by a large class of conical expanders. As an application, we remove the positive curvature condition in a classification result by [Cho14] in dimension three and show that any 3-dimensional gradient Ricci expander C 2 asymptotic to C(S 2 ), dt 2 + αt 2 g S 2 is rotationally symmetric, where α ∈ (0, 1] is a constant and g S 2 is the standard metric on S 2 with constant curvature 1.

Volume Growth Estimates of Gradient Ricci Solitons

“…Remark 3.9 Recently, the authors [22] have showed that (3.9) holds on any fourdimensional complete non-Ricci-flat steady soliton singularity model (see also [12,15,18,51]).…”

On Curvature Estimates for four-dimensional gradient Ricci solitons

“…Remark 1.1. Curvature estimates in the form of |Rm| ≤ CR were first derived by Munteanu-Wang [65] for 4-dimensional gradient shrinking Ricci solitons and subsequently obtained for 4-dimensional gradient steady solitons in [27], [18, Theorem 5.2], [36] and [28]; see also related results in [23] for 4D Ricci shrinkers and [22] for 4D Ricci expanders.…”