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 $3$-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.
- Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684
- John A. Buckland, Short-time existence of solutions to the cross curvature flow on 3-manifolds, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1803–1807. MR 2207496, DOI https://doi.org/10.1090/S0002-9939-05-08204-3
- Bennett Chow and Richard S. Hamilton, The cross curvature flow of 3-manifolds with negative sectional curvature, Turkish J. Math. 28 (2004), no. 1, 1–10. MR 2055396
- Cao, Xiaodong; Ni, Yilong; Saloff-Coste, Laurent. Cross curvature flow on locally homogeneous three-manifolds. arXiV:0708.1922.
- F. T. Farrell and P. Ontaneda, A caveat on the convergence of the Ricci flow for pinched negatively curved manifolds, Asian J. Math. 9 (2005), no. 3, 401–406. MR 2214959, DOI https://doi.org/10.4310/AJM.2005.v9.n3.a6
- Farrell, F. Thomas and Ontaneda, Pedro. On the topology of the space of negatively curved metrics. arXiV:math.DG/0607367.
- Glickenstein, David. Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flows. arXiv:0710.1276.
- M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12. MR 892185, DOI https://doi.org/10.1007/BF01404671
- Christine Guenther, James Isenberg, and Dan Knopf, Stability of the Ricci flow at Ricci-flat metrics, Comm. Anal. Geom. 10 (2002), no. 4, 741–777. MR 1925501, DOI https://doi.org/10.4310/CAG.2002.v10.n4.a4
- Norihito Koiso, On the second derivative of the total scalar curvature, Osaka Math. J. 16 (1979), no. 2, 413–421. MR 539596
- Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR 1329547
- Li Ma and Dezhong Chen, Examples for cross curvature flow on 3-manifolds, Calc. Var. Partial Differential Equations 26 (2006), no. 2, 227–243. MR 2222245, DOI https://doi.org/10.1007/s00526-005-0366-1
- Rugang Ye, Ricci flow, Einstein metrics and space forms, Trans. Amer. Math. Soc. 338 (1993), no. 2, 871–896. MR 1108615, DOI https://doi.org/10.1090/S0002-9947-1993-1108615-3
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C44, 53C21, 58J35
Retrieve articles in all journals with MSC (2000): 53C44, 53C21, 58J35
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
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
