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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic stability of the cross curvature flow at a hyperbolic metric


Authors: Dan Knopf and Andrea Young
Journal: Proc. Amer. Math. Soc. 137 (2009), 699-709
MSC (2000): Primary 53C44, 53C21, 58J35
Published electronically: September 3, 2008
MathSciNet review: 2448593
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Abstract: We show that for any hyperbolic metric on a closed $ 3$-manifold, there exists a neighborhood such that every solution of a normalized cross curvature flow with initial data in this neighborhood exists for all time and converges to a constant-curvature metric. We demonstrate that the same technique proves an analogous result for Ricci flow. Additionally, we prove short-time existence and uniqueness of cross curvature flow under slightly weaker regularity hypotheses than were previously known.


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Additional Information

Dan Knopf
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78713
Email: danknopf@math.utexas.edu

Andrea Young
Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78713
Address at time of publication: Department of Mathematics, The University of Arizona, Tucson, Arizona 85721
Email: ayoung@math.arizona.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09534-8
PII: S 0002-9939(08)09534-8
Keywords: Cross curvature flow, asymptotic stability, hyperbolic metrics
Received by editor(s): July 16, 2007
Received by editor(s) in revised form: February 5, 2008
Published electronically: September 3, 2008
Additional Notes: The first author was partially supported by NSF grants DMS-0505920 and DMS-0545984.
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2008 by the authors