Some rigidity results on shrinking gradient Ricci soliton

TL;DR

This paper establishes new rigidity results for complete shrinking gradient Ricci solitons under natural conditions, including a classification result for solitons with constant scalar curvature, using maximum principle techniques.

Contribution

It generalizes previous rigidity results and provides a new proof for the classification of certain shrinking gradient Ricci solitons with constant scalar curvature.

Findings

Shrinking gradient Ricci soliton with scalar curvature R=1 is isometric to a finite quotient of ^2 t2 t2.

Provides a new proof of Cheng-Zhou's classification result.

Extends rigidity results under natural geometric conditions.

Abstract

Suppose $(M^n, g, f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that shrinking gradient Ricci soliton with constant scalar curvature $R=1$ is isometric to a finite quotient of $\mathbb{R}^2\times \mathbb{S}^2$, giving a new proof of the main results of Cheng-Zhou \cite{Cheng-Zhou}.