Steady Ricci solitons with horizontally $\epsilon$-pinched Ricci curvature

TL;DR

This paper proves that certain noncollapsed steady Ricci solitons with specific curvature conditions are rotationally symmetric, extending understanding of their geometric structure and symmetry properties.

Contribution

It establishes rotational symmetry for a class of noncollapsed steady Ricci solitons under horizontally $ ext{ extepsilon}$-pinched Ricci curvature and additional scalar curvature conditions.

Findings

Noncollapsed steady Ricci solitons with horizontally $ ext{ extepsilon}$-pinched Ricci curvature are rotationally symmetric.

Rotational symmetry holds for solitons with a unique equilibrium point and specific scalar curvature growth.

The results extend symmetry classification in Ricci flow geometry.

Abstract

In this paper, we prove that any $\kappa$-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally $\epsilon$-pinched Ricci curvature must be rotationally symmetric. As an application, we show that any $\kappa$-noncollapsed gradient steady Ricci soliton $(M^n, g,f)$ with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature $R(x)$ satisfies $\lim_{r(x)\rightarrow\infty}R(x)f(x)=C_0\sup_{x\in M}R(x)$ with $C_0>\frac{n-2}{2}$.