Asymptotic stability of the cross curvature flow at a hyperbolic metric
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- by Dan Knopf and Andrea Young PDF
- Proc. Amer. Math. Soc. 137 (2009), 699-709
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.References
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Additional Information
- Dan Knopf
- Affiliation: Department of Mathematics, University of Texas at Austin, Austin, Texas 78713
- MR Author ID: 661950
- 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
- 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
- © Copyright 2008 by the authors
- Journal: Proc. Amer. Math. Soc. 137 (2009), 699-709
- MSC (2000): Primary 53C44, 53C21, 58J35
- DOI: https://doi.org/10.1090/S0002-9939-08-09534-8
- MathSciNet review: 2448593