Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flows

TL;DR

This paper uses Riemannian groupoids to analyze the long-term behavior of three-dimensional homogeneous solutions in Ricci and cross curvature flows, identifying soliton solutions and flow limits.

Contribution

It applies the Riemannian groupoid method to study collapsing solutions and finds new soliton metrics for cross curvature flow on specific geometries.

Findings

Identified cross curvature solitons on Sol and Nil.

Showed flow of SL(2,R) converges to Sol.

Demonstrated long-term behavior of solutions using groupoids.

Abstract

In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the limit. We also review Lott's results on the long term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross curvature soliton metrics on Sol and Nil, and show that the cross curvature flow of SL(2,R) limits to Sol.