Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

TL;DR

This paper proves that five-dimensional complete noncompact gradient shrinking Ricci solitons with constant scalar curvature are finite quotients of 23, establishing a rigidity result in geometric analysis.

Contribution

It classifies five-dimensional gradient shrinking Ricci solitons with constant scalar curvature, showing they are finite quotients of 23, thus extending rigidity results in Ricci flow theory.

Findings

Such solitons are finite quotients of 23

The scalar curvature condition $R=3 \, \lambda$ is crucial

The result characterizes the geometry of these solitons

Abstract

Let $(M, g, f)$ be a $5$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \lambda g$, where $\text{Ric}$ is the Ricci tensor and $\nabla^2f$ is the Hessian of the potential function $f$. We prove that it is a finite quotient of $\mathbb{R}^2\times \mathbb{S}^3$ if $M$ has constant scalar curvature $R=3 \lambda$.