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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.



References

  1. Baird P., Danielo L. Three-dimensional Ricci solitons which project to surfaces. Journal fur die Reine und Angewandte Mathematik (2007) 608:65–91.[ISI]
  2. Bridson M. R., Haefliger A. Metric Spaces of Non-Positive Curvature (1999) Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319. Berlin: Springer.
  3. Buckland J. A. Short-time existence of solutions to the cross curvature flow on 3-manifolds. Proceedings of the American Mathematical Society (2006) 134(6):1803–7.[CrossRef][ISI]
  4. Buser P., Karcher H. Gromov's Almost Flat Manifolds (1981) Astérisque, 81. Société Mathématique de France, Paris.
  5. Cao X., Ni Y., Saloff-Coste L. Cross curvature flow on locally homogeneous three manifolds (I). (2007) preprint arXiv:0708.1922v1.
  6. Cao H.-D., Zhu X.-P. A complete proof of the Poincaré and geometrization conjectures—Application of the Hamilton–Perelman theory of the Ricci flow. The Asian Journal of Mathematics (2006) 10(2):165–492.
  7. Chow B., Chu S.-C., Glickenstein D., Guenther C., Isenberg J., Ivey T., Knopf D., Lu P., Luo F., Ni L. The Ricci Flow: Techniques and Applications. Part 1. Geometric Aspects. (2007) Mathematical Surveys and Monographs, 135. Providence, RI: American Mathematical Society.
  8. Chow B., Hamilton R. S. The cross curvature flow of 3-manifolds with negative sectional curvature. Turkish Journal of Mathematics (2004) 28(1):1–10.
  9. Chow B., Knopf D. The Ricci Flow: An Introduction (2004) 325 pp. Mathematical Surveys and Monographs, 110. Providence, RI: American Mathematical Society.
  10. Chow B., Lu P., Ni L. Hamilton's Ricci Flow (2006) 608 pp. Graduate Studies in Mathematics, 77. Providence, RI: American Mathematical Society, Science Press, New York.
  11. Fukaya K. A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters. Journal of Differential Geometry (1988) 28(1):1–21.[ISI]
  12. Glickenstein D. Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates. Geometry and Topology (2003) 7:487–510. (electronic).[CrossRef]
  13. Gromov M. Almost flat manifolds. Journal of Differential Geometry (1978) 13(2):231–41.
  14. Guenther C., Isenberg J., Knopf D. Linear stability of homogeneous Ricci solitons. International Mathematics Research Notices (2006) Art. ID 96253, 30 pp.
  15. Haefliger A. "Leaf closures in Riemannian foliations." I. A fête of Topology (1988) 3–32. Boston, MA: Academic Press.
  16. Hamilton R. S. Three-manifolds with positive Ricci curvature. Journal of Differential Geometry (1982) 17(2):255–306.[ISI]
  17. Hamilton R. S. The Formation of Singularities in the Ricci Flow (1995) Surveys in Differential Geometry, Vol. II (Cambridge, MA, 1993), 7–136. Cambridge, MA: International Press.
  18. Hamilton R. S. A compactness property for solutions of the Ricci flow. American Journal of Mathematics (1995) 117(3):545–72.[CrossRef][ISI]
  19. Isenberg J., Jackson M. Ricci flow of locally homogeneous geometries on closed manifolds. Journal of Differential Geometry (1992) 35(3):723–41.[ISI]
  20. Kleiner B., Lott J. Notes on Perelman's papers. (2006) preprint arXiv:math/0605667v2.
  21. Knopf D., McLeod K. Quasi-convergence of model geometries under the Ricci flow. Communications in Analysis and Geometry (2001) 9(4):879–919.[ISI]
  22. Knopf D., Young A. Asymptotic stability of the cross curvature flow at a hyperbolic metric. In: Proceedings of the American Mathematical Society (2007) preprint arXiv:math/0609767v2.
  23. Lott J. On the long-time behavior of type III Ricci flow solutions. Mathematische Annalen (2007) 339:627–66.[CrossRef][ISI]
  24. Lu P. A compactness property for solutions of the Ricci flow on orbifolds. American Journal of Mathematics (2001) 123(6):1103–34.[CrossRef][ISI]
  25. Mackenzie K. C. H. General Theory of Lie Groupoids and Lie Algebroids (2005) 501 pp. London Mathematical Society Lecture Note Series, 213. Cambridge: Cambridge University Press.
  26. Ma L., Chen D. Examples for cross curvature flow on 3-manifolds. Calculus of Variations and Partial Differential Equations (2006) 26(2):227–43.[CrossRef][ISI]
  27. Moerdijk I., Mrcun J. Introduction to Foliations and Lie Groupoids (2003) 173 pp. Cambridge Studies in Advanced Mathematics, 91. Cambridge: Cambridge University Press.
  28. Morgan J., Tian G. Ricci Flow and the Poincaré Conjecture (2007) 521 pp. Clay Mathematics Monographs, 3. Providence, RI: American Mathematical Society, Clay Mathematics Institute, Cambridge, MA.
  29. Perelman G. The entropy formula for the Ricci flow and its geometric applications. (2002) preprint arXiv:math/0211159v1.
  30. Perelman G. Ricci flow with surgery on three-manifolds. (2003) preprint arXiv:math/0303109v1.
  31. Salem É. Une généralisation du théorème de Myers–Steenrod aux pseudogroupes d'isométries. (1988) 38(2):185–200. (French) [A generalization of the Myers–Steenrod theorem to pseudogroups of isometries]. Universit¥e de Grenoble. Annales de lí;Institut Fourier.
  32. Scott P. The geometries of 3-manifolds. The Bulletin of the London Mathematical Society (1983) 15(5):401–87.[CrossRef]

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This Article
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