Cohomogeneity one Ricci Solitons from Hopf Fibrations

TL;DR

This paper constructs new cohomogeneity one Ricci solitons on non-trivial bundles using Hopf fibrations, extending the understanding of Ricci solitons and Einstein metrics with positive scalar curvature.

Contribution

It demonstrates the existence of parameter families of non-homothetic complete steady and expanding Ricci solitons under specific isotropy conditions, linking numerical detection to rigorous analysis.

Findings

Existence of non-homothetic Ricci solitons on non-trivial bundles.

Connection between Ricci solitons and Einstein metrics of positive scalar curvature.

Application of techniques to m-quasi-Einstein metrics.

Abstract

This paper studies cohomogeneity one Ricci solitons. If the isotropy representation of the principal orbit $G/K$ consists of two inequivalent $Ad_K$-invariant irreducible summands, the existence of parameter families of non-homothetic complete steady and expanding Ricci solitons on non-trivial bundles is shown. These examples were detected numerically by Buzano-Dancer-Gallaugher-Wang. The analysis of the corresponding Ricci flat trajectories is used to reconstruct Einstein metrics of positive scalar curvature due to B\"ohm. The techniques also apply to $m$-quasi-Einstein metrics.