A comparison theorem for steady Ricci solitons

TL;DR

This paper characterizes steady Ricci solitons under certain geometric conditions, showing they are either flat, product, or Bryant solitons, and establishes conditions for uniqueness and flatness.

Contribution

It provides a comparison theorem classifying steady Ricci solitons with specific geometric conditions and curvature bounds, including new uniqueness results for the Bryant soliton.

Findings

Steady Ricci solitons are either Ricci flat, product, or Bryant solitons under given conditions.

Complete non-compact steady Ricci solitons with controlled curvature are Bryant solitons.

Positively pinched Ricci curvature implies Ricci flatness for steady solitons.

Abstract

We prove that a steady gradient Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton $N^{n-1}\times\mathbb{R}$, where $N^{n-1}$ is Ricci flat, or isometric to the Bryant soliton (up to scalings), provided that a couple of geometric conditions inspired by the cigar soliton hold. As an application, we prove that any complete non-compact steady Ricci soliton with positive Ricci curvature controlled by the scalar curvature $R$, curvature tensor $Rm$ satisfying $|Rm|r\to o(1)$ and $R\to\infty$, as $r\to\infty$, must be the Bryant soliton. Moreover, we prove that any complete steady soliton with positively pinched Ricci curvature must be Ricci flat.