Riemannian Groupoids and Solitons for Three-Dimensional Homogeneous Ricci and Cross-Curvature Flows

In this paper, we investigate the behavior of three-dimensional homogeneous solutions of the cross-curvature flow using Riemannian groupoids. The Riemannian groupoid technique, originally introduced by J. Lott, allows us to investigate the long-term behavior of collapsing solutions of the flow, prod...

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Bibliographic Details
Published in:International mathematics research notices 2008-01, Vol.2008
Main Author: Glickenstein, David
Format: Article
Language:English
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Summary:In this paper, we investigate the behavior of three-dimensional homogeneous solutions of the cross-curvature flow using Riemannian groupoids. The Riemannian groupoid technique, originally introduced by J. Lott, allows us to investigate the long-term behavior of collapsing solutions of the flow, producing solitons 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.
ISSN:1073-7928
1687-0247
DOI:10.1093/imrn/rnn034