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International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn034, 49 pages, doi:10.1093/imrn/rnn034 published on April 25, 2008
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© The Author 2008. Published by Oxford University Press. All rights reserved. For Permissions, please e-mail: journals.permissions@oxfordjournals.org

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

David Glickenstein

Department of Mathematics, University of Arizona, 617 N Santa Rita, P.O. Box 210089, Tucson, AZ 85721, USA

Correspondence: Correspondence to be sent to: glickenstein{at}math.arizona.edu

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.


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