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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
DOI: https://doi.org/10.1090/S0002-9939-05-08204-3
Published electronically: December 16, 2005
MathSciNet review: 2207496
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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. B. Chow, Deforming convex hypersurfaces by the n-th root of the Gaussian curvature, J. Diff. Geom., 22 (1985), 117-138. MR 0826427 (87f:58155)
  • 2. B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math., 87 (1987), 63-82. MR 0862712 (88a:58204)
  • 3. B. Chow and R. Hamilton, The cross curvature flow of 3-manifolds with negative sectional curvature, Turk. J. Math., 28 (2004), 1-10. MR 2055396 (2005a:53107)
  • 4. D. DeTurck, Deforming metrics in the direction of their Ricci tensors, J. Diff. Geom., 18 (1983), 157-162. MR 0697987 (85j:53050)
  • 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. Hamilton, The heat equation shrinking convex plane curves, J. Diff. Geom., 23 (1986), 69-96. MR 0840401 (87m:53003)
  • 7. R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom., 17 (1982), 255-306. MR 0664497 (84a:53050)

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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: https://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.

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