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

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.

**1.**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****2.**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 (electronic). MR**2207496**, 10.1090/S0002-9939-05-08204-3**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****4.****Cao, Xiaodong; Ni, Yilong; Saloff-Coste, Laurent**. Cross curvature flow on locally homogeneous three-manifolds.`arXiV:0708.1922`.**5.**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**, 10.4310/AJM.2005.v9.n3.a6**6.****Farrell, F. Thomas and Ontaneda, Pedro.**On the topology of the space of negatively curved metrics.`arXiV:math.DG/0607367`.**7.****Glickenstein, David**. Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross curvature flows.`arXiv:0710.1276`.**8.**M. Gromov and W. Thurston,*Pinching constants for hyperbolic manifolds*, Invent. Math.**89**(1987), no. 1, 1–12. MR**892185**, 10.1007/BF01404671**9.**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**, 10.4310/CAG.2002.v10.n4.a4**10.**Norihito Koiso,*On the second derivative of the total scalar curvature*, Osaka J. Math.**16**(1979), no. 2, 413–421. MR**539596****11.**Alessandra Lunardi,*Analytic semigroups and optimal regularity in parabolic problems*, Progress in Nonlinear Differential Equations and their Applications, 16, Birkhäuser Verlag, Basel, 1995. MR**1329547****12.**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**, 10.1007/s00526-005-0366-1**13.**Rugang Ye,*Ricci flow, Einstein metrics and space forms*, Trans. Amer. Math. Soc.**338**(1993), no. 2, 871–896. MR**1108615**, 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

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

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