Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Width and flow of hypersurfaces by curvature functions


Authors: Maria Calle, Stephen J. Kleene and Joel Kramer
Journal: Trans. Amer. Math. Soc. 363 (2011), 1125-1135
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/S0002-9947-2010-05057-3
Published electronically: October 14, 2010
MathSciNet review: 2737259
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we generalize a well-known estimate of Colding-Minicozzi for the extinction time of convex hypersurfaces in Euclidean space evolving by their mean curvature to a much broader class of parabolic curvature flows.


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

  • 1. B. Andrews. Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994), no. 2, 151-171. MR 1385524 (97b:53012)
  • 2. B. Andrews. Moving surfaces by non-concave curvature functions. arXiv:math math.DG/0402273
  • 3. B. Andrews. Flow of hypersurfaces by curvature functions. Workshop on Theoretical and Numerical Aspects of Geometric Variational Problems (Canberra, 1990), 1-10, Proc. Centre Math. Appl. Austral. Nat. Univ. 26, Austral. Nat. Univ., Canberra, 1991. MR 1139026 (92k:53010)
  • 4. B. Andrews. Pinching estimates and motion of hypersurfaces by curvature functions. math.DG/0402311v1
  • 5. B. Andrews. Volume-preserving anisotropic mean curvature flow, Indiana Univ. Math. J. 50 (2001), no. 2, 783-827. MR 1871390 (2002m:53105)
  • 6. K. Borsuk. Sur la courbure totale des courbes fermes, Ann. Soc. Polon. Math. 20 (1947), 251-265. MR 0025757 (10:60e)
  • 7. T. H. Colding and W. P. Minicozzi. Width and mean curvature flow. math.DG/0705.3827v2
  • 8. T. H. Colding and W. P. Minicozzi. Width and finite extinction time of Ricci Flow. math.DG/0707.0108v1
  • 9. T. H. Colding and W. P. Minicozzi. Estimates for the extinction time for the Ricci Flow on certain $ 3$-manifolds and a question of Perelman. arXiv:math/0308090v3
  • 10. W. Fenchel. Über Krümmung und Windung geschlossener Raumkurven, Math. Ann. 101 (1929), no. 1, 238-252. MR 1512528
  • 11. G. Huisken and A. Polden. Geometric evolution equations for hypersurfaces. Calculus of variations and geometric evolution problems (Cetraro, 1996), 45-84, Lecture Notes in Math., 1713, Springer, Berlin, 1999. MR 1731639 (2000j:53090)
  • 12. F. Schulze. Evolution of convex hypersurfaces by powers of the mean curvature. Math. Z. 251 (2005), no. 4, 721-733. MR 2190140 (2006h:53065)
  • 13. L. Simon. Lectures on geometric measure theory, Australian National University Centre for Mathematical Analysis, Canberra, 1983. MR 756417 (87a:49001)
  • 14. L. Simon. Existence of surfaces minimizing the Willmore functional, CAG 1 (1993) 281-326. MR 1243525 (94k:58028)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C44

Retrieve articles in all journals with MSC (2010): 53C44


Additional Information

Maria Calle
Affiliation: Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Golm, Germany
Email: maria.calle@aei.mpg.de

Stephen J. Kleene
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
Address at time of publication: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139
Email: skleene@math.jhu.edu, skleene@math.mit.edu

Joel Kramer
Affiliation: Department of Mathematics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland 21218
Email: jkramer@math.jhu.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05057-3
Received by editor(s): April 18, 2008
Published electronically: October 14, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society