Complete steady gradient Yamabe solitons with positive scalar curvature are rotationally symmetric

TL;DR

The paper proves that any nontrivial complete steady gradient Yamabe soliton with positive scalar curvature is rotationally symmetric in all dimensions, solving the Yamabe soliton version of the Perelman conjecture. It leverages Tashiro's classification for manifolds with $\nabla\nabla F=\varphi g$ to reduce the geometry to a warped product or rotationally symmetric form and derives an autonomous ODE for $\rho=F'$. By showing $\bar R>0$ and that $\rho$ is strictly increasing and unbounded, it rules out nonradial structures and confirms rotational symmetry as the only possibility. This result extends prior work that required locally conformally flat or specific curvature conditions and completes the higher-dimensional understanding of steady Yamabe solitons as models for singularity formation.

Abstract

In this paper, we solve the Yamabe soliton version of the Perelman conjecture. We show that any nontrivial complete steady gradient Yamabe solitons with positive scalar curvature are rotationally symmetric.

Theorems & Definitions (7)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1: Tashiro65
• Remark 2.2
• proof : Proof of Theorem \ref{['main']}
• Theorem 2.3
• Remark 2.4