Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Short-time existence of solutions to the cross curvature flow on 3-manifolds


Author: John A. Buckland
Journal: Proc. Amer. Math. Soc. 134 (2006), 1803-1807
MSC (2000): Primary 53C44, 35K55
Published electronically: December 16, 2005
MathSciNet review: 2207496
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a compact 3-manifold with an initial Riemannian metric of positive (or negative) sectional curvature, we prove the short-time existence of a solution to the cross curvature flow. This is achieved using an idea first introduced by DeTurck (1983) in his work establishing the short-time existence of solutions to the Ricci flow.


References [Enhancements On Off] (What's this?)

  • 1. Bennett Chow, Deforming convex hypersurfaces by the 𝑛th root of the Gaussian curvature, J. Differential Geom. 22 (1985), no. 1, 117–138. MR 826427
  • 2. Bennett Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987), no. 1, 63–82. MR 862712, 10.1007/BF01389153
  • 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. Dennis M. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Differential Geom. 18 (1983), no. 1, 157–162. MR 697987
  • 5. D. DeTurck, Deforming metrics in the direction of their Ricci tensors (Improved version), in Collected Papers on Ricci Flow, H.D. Cao, B. Chow, S.C. Chu and S.T. Yau, editors, Series in Geometry and Topology 37 Int. Press (2003), 163-165.
  • 6. M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geom. 23 (1986), no. 1, 69–96. MR 840401
  • 7. Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255–306. MR 664497

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C44, 35K55

Retrieve articles in all journals with MSC (2000): 53C44, 35K55


Additional Information

John A. Buckland
Affiliation: Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
Email: John.Buckland@maths.anu.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08204-3
Keywords: Nonlinear evolution equations, curvature flow, short-time existence
Received by editor(s): January 31, 2005
Received by editor(s) in revised form: February 1, 2005
Published electronically: December 16, 2005
Additional Notes: This research was partially supported by an Australian Research Council Discovery grant entitled Geometric evolution equations and global effects of curvature
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.