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)



A new flow on starlike curves in $\mathbb{R}^3$

Authors: Rongpei Huang and David A. Singer
Journal: Proc. Amer. Math. Soc. 130 (2002), 2725-2735
MSC (2000): Primary 53A04; Secondary 53A15
Published electronically: April 11, 2002
MathSciNet review: 1900890
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note we find a new evolution equation for starlike curves in $\mathbb{R}^3$. We study the evolution of the subaffine curvature and subaffine torsion under the flow and show that it is completely integrable. The solutions to the evolution which move without changing affine shape are subaffine elastic curves. We integrate the subaffine elastica by quadratures.

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

  • 1. G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, New York, 1967. MR 2000j:76001 (Review of second paperback edition)
  • 2. G.Boole, A Treatise on Differential Equations, G.E. Stechert & Co., New York, 1931. MR 21:5760 (Review of 5th edition)
  • 3. R.Bryant & P.Griffiths, Reduction for constrained variational problem and $\int k^2/2ds$, Amer. J. Math. 108 (1986), 525-570. MR 88a:58044
  • 4. P.Byrd & M.Friedman, Handbook of elliptic integrals for engineers and physicists, Springer-Verlag, Berlin, 1954. MR 15:702a
  • 5. E. Calabi, P. Olver, & A. Tannenbaum, Affine geometry, curve flows, and invariant numerical approximations, Advances in Mathematics 124 (1996), 154-196. MR 97k:58040
  • 6. P. Giblin & G. Sapiro, Affine-invariant distances, envelopes, and symmetry sets, Geometriae Dedicata 71 (1998), 237-261. MR 99e:53002
  • 7. H.Goldstine, A history of the calculus of variations from the 17th through the 19th century, Springer-Verlag, New York, 1980. MR 83i:01036
  • 8. H.Hasimoto, A soliton on a vortex filament, J. Fluid Mech. 51 (1972), 477-486.
  • 9. R. Huang, Affine and Subaffine Elastic Curves in $\mathbb{R}^2$ and $\mathbb{R}^3$, thesis, Case Western Reserve University, 1999.
  • 10. G.L. Lamb, Elements of Soliton Theory, Wiley Interscience, New York, 1980. MR 82f:35165
  • 11. J.Langer & R.Perline, Poisson geometry of the filament equation, J. Nonlinear Science 1 (1991), 71-93. MR 92k:58118
  • 12. -, Local geometric invariants of integrable evolution equations, J. Math. Phys. 35(4) (1994), 1732-1737. MR 95c:58095
  • 13. J.Langer & D.Singer, The total squared curvature of closed curves, J. Differential Geometry, 20 (1984) 1-22. MR 86i:58030
  • 14. -, Knotted elastic curves in $\mathbb{R}^3$, J. London Math. Soc. 16 (1984), 512-618. MR 87d:53004
  • 15. -, Liouville integrability of geometric variational problems, Comment. Math. Helvetici 69 (1994), 272-280. MR 95f:58042
  • 16. -, Lagrangian aspects of the Kirchhoff elastic rod, SIAM Review 38 (1996), 605-618. MR 97h:73050
  • 17. J.Marsden & A.Weinstein, Coadjoint orbits,vortices and clebsch variables for incompressible fluids, Physica 7D(1983), 305-323. MR 85g:58039
  • 18. K.Nomizu & T.Sasaki, Affine differential geometry, Cambridge University Press, 1994. MR 96e:53014
  • 19. S.Novikov, Solitons and geometry, Accademia Nazionale Deilincei, 1992. MR 95i:58091
  • 20. U. Pinkall, Hamiltonian flows on the space of star-shaped curves, Results Math. 27 (1995), 328-332. MR 96i:58078
  • 21. B.Su, Affine differential geometry, Beijing, 1983. MR 85g:53010

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53A04, 53A15

Retrieve articles in all journals with MSC (2000): 53A04, 53A15

Additional Information

Rongpei Huang
Affiliation: Department of Mathematics, East China Normal University, Shanghai, 200062, People’s Republic of China

David A. Singer
Affiliation: Department of Mathematics, Case Western Reserve University, Cleveland, Ohio 44106-7058

Received by editor(s): February 4, 2000
Published electronically: April 11, 2002
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society