On rotationally invariant shrinking gradient Ricci solitons

TL;DR

This paper classifies all rotationally symmetric gradient Ricci shrinking solitons in higher dimensions, showing that only standard geometries like round spheres, flat spaces, and cylinders satisfy the conditions.

Contribution

It proves the uniqueness of complete rotationally symmetric gradient Ricci shrinking solitons on certain manifolds, extending the understanding of these geometric structures.

Findings

Complete examples on $S^n$ are round spheres.

Complete examples on $ ^n$ are flat spaces.

Complete examples on $ imes S^{n-1}$ are standard cylinders.

Abstract

In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are, respectively, the round, flat, and standard cylindrical metrics.