Hamilton-Ivey estimates for gradient Ricci solitons

TL;DR

This paper establishes curvature estimates for 4-dimensional steady gradient Ricci solitons and provides conditions for positive curvature in 3-dimensional expanders, leading to classification results and symmetry conclusions.

Contribution

It introduces new curvature bounds for 4D steady Ricci solitons and removes positive curvature assumptions in 3D expanders, advancing classification and symmetry results.

Findings

Any 4D non-Ricci-flat steady gradient Ricci soliton satisfies |Rm| ≤ cR.

A lower bound for the curvature operator in 4D steady gradient solitons with linear scalar decay.

3D expanding gradient Ricci solitons asymptotic to certain cones are rotationally symmetric.

Abstract

We first show that any $4$-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy $|Rm|\leq cR$ 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 Chodosh 14 in dimension three and show that any $3$-dimensional gradient Ricci expander $C^2$ asymptotic to $\left(C(\mathbb S^2), dt^2+\alpha t^2 g_{\mathbb{S}^2}\right)$ is rotationally symmetric, where $\alpha \in (0,1]$ is a constant and $g_{\mathbb{S}^2}$ is the standard metric on $\mathbb{S}^2$ with constant curvature $1$.