Advertisement

Annals of Global Analysis and Geometry

, Volume 43, Issue 4, pp 385–395 | Cite as

The energy functional on the Virasoro–Bott group with the L 2-metric has no local minima

  • Martins BruverisEmail author
Article

Abstract

The geodesic equation for the right invariant L 2-metric (which is a weak Riemannian metric) on each Virasoro–Bott group is equivalent to the KdV-equation. We prove that the corresponding energy functional, when restricted to paths with fixed endpoints, has no local minima. In particular, solutions of KdV do not define locally length-minimizing paths.

Keywords

Diffeomorphism group Virasoro group Geodesic distance 

Mathematics Subject Classification (2000)

Primary 35Q53 58B20 58D05 58D15 58E12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnold V.I.: Sur la géometrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier 16, 319–361 (1966)CrossRefGoogle Scholar
  2. 2.
    Bauer M., Bruveris M., Harms P., Michor P.: Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation. Ann. Global Anal. Geom. 41, 461–472 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Constantin A., Kappeler T., Kolev B., Topalov P: On geodesic exponential maps of the Virasoro group. Ann. Global Anal. Geom. 31(2), 155–180 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Constantin A., Kolev B.: Geodesic flow on the diffeomorphism group of the circle. Comment. Math. Helv. 78(4), 787–804 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Ebin D.G., Marsden J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. (2) 92, 102–163 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gay-Balmaz F.: Well-posedness of higher dimensional Camassa–Holm equations. Bull. Transilv. Univ. Braşov Ser. III 2(51), 55–58 (2009)MathSciNetGoogle Scholar
  7. 7.
    Khesin B., Wendt R.: The geometry of infinite-dimensional groups. Springer, New York (2009)Google Scholar
  8. 8.
    Kriegl A., Michor P.W.: The Convenient Setting of Global Analysis, volume 53 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (1997)Google Scholar
  9. 9.
    Michor, P.W.: Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach. In: Bove, A., Colombini, F., Del Santo, D. (eds.) Phase Space Analysis of Partial Differential Equations, volume 69 of Progress in Nonlinear Differential Equations and their Applications, pp. 133–215. Birkhäuser, Boston (2006)Google Scholar
  10. 10.
    Michor, P.W., Mumford, D.: Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Doc. Math. 10, 217–245 (electronic) (2005)Google Scholar
  11. 11.
    Michor P.W., Ratiu T.: Geometry of the Virasoro–Bott group. J. Lie Theory 8, 293–309 (1998)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Misiołek G.: Conjugate points in the Bott–Virasoro group and the KdV equation. Proc. Amer. Math. Soc. 125(3), 935–940 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ovsienko V.Y., Khesin B.A.: Korteweg–de Vries superequations as an Euler equation. Funct. Anal. Appl. 21, 329–331 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Segal G.: The geometry of the KdV equation. Topological methods in quantum field theory (Trieste, 1990). Internat. J. Modern Phys. A 6(16), 2859–2869 (1991)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of MathematicsImperial College LondonLondonUK

Personalised recommendations