On curvature estimates for four-dimensional gradient Ricci solitons

TL;DR

This survey analyzes curvature control for four-dimensional gradient Ricci solitons across shrinking, expanding, and steady types, highlighting both known results and new sharp bounds. It surveys MW’s shrinker estimates $|\mathrm{Rm}| \le C R$ and related derivatives, and discusses key tools like the operator $\Delta_f$ and maximum-principle arguments, as well as cone-type asymptotics when $R\to0$ at infinity. For expanding solitons, it presents Cao–Liu’s almost-sharp bounds $|\mathrm{Rm}| \le C/(1-a) R^a$ (and $|\nabla \mathrm{Rm}|$) for $0\le a<1$, with implications for finite asymptotic curvature ratio and asymptotic cones, while noting the fundamental sign difference in the governing equations. In steady solitons, sharp results $|\mathrm{Rm}| \le C R$ (and $|\nabla \mathrm{Rm}| \le C R$) under $Rc>0$ and $R$ attaining a maximum extend the shrinker toolkit to the steady regime. The paper closes with open questions on lower bounds for $R$ in expanders, gradient-structure estimates, and the reach of sharp curvature bounds in singularity-models, guiding future directions for the geometry of 4D solitons.

Abstract

In this survey paper, we analyse and compare the recent curvature estimates for three types of $4$-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature estimates for $4$-dimensional gradient steady Ricci solitons, including the sharp curvature estimate $|Rm|\le C R$ for gradient steady Ricci solitons with positive Ricci curvature (see Theorem 1.1).

Theorems & Definitions (42)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 2.1
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Proposition 3.1