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
DOI:
https://doi.org/10.1090/S0002-9939-08-09534-8
Published electronically:
September 3, 2008
MathSciNet review:
2448593
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We show that for any hyperbolic metric on a closed 
-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:
https://doi.org/10.1090/S0002-9939-08-09534-8
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
