A note on four dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

TL;DR

This paper provides an alternative proof that four-dimensional complete noncompact gradient shrinking Ricci solitons with scalar curvature 1 are finite quotients of , by analyzing their asymptotic geometry.

Contribution

It offers a new proof approach for classifying four-dimensional shrinking Ricci solitons with constant scalar curvature, focusing on their asymptotic geometry.

Findings

Confirmed that such solitons are finite quotients of .

Provided an alternative proof method based on asymptotic analysis.

Reinforced the classification result for these Ricci solitons.

Abstract

Let $(M^4, g, f)$ be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. If its scalar curvature is $1$, Cheng-Zhou \cite{Cheng-Zhou} proved that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^2$. In this note we present an alternative proof by analyzing the asymptotic geometry at infinity.