Cohomogeneity one shrinking Ricci solitons: an analytic and numerical study

TL;DR

This paper combines analytical and numerical techniques to study cohomogeneity one shrinking Ricci solitons, revealing a winding number around Einstein solutions, non-existence results, and numerical insights into specific orbit types.

Contribution

It introduces a new winding number concept, proves non-existence results for certain orbits, and provides numerical analysis for specific cases in cohomogeneity one Ricci solitons.

Findings

Existence of a winding number around Einstein solutions

Non-existence results for certain orbit types

Numerical investigations of selected orbit types

Abstract

We use analytical and numerical methods to investigate the equations for cohomogeneity one shrinking gradient Ricci solitons. We show the existence of a winding number for this system around the subvariety of phase space corresponding to Einstein solutions and obtain some estimates for it. We prove a non-existence result for certain orbit types, analogous to that of Bohm in the Einstein case. We also carry out numerical investigations for selected orbit types.