Archive for Rational Mechanics and Analysis

, Volume 229, Issue 3, pp 1015–1035 | Cite as

The Exponential Map of the Group of Area-Preserving Diffeomorphisms of a Surface with Boundary

  • James Benn
  • Gerard Misiołek
  • Stephen C. Preston


We prove that the Riemannian exponential map of the right-invariant L2 metric on the group of volume-preserving diffeomorphisms of a two-dimensional manifold with a nonempty boundary is a nonlinear Fredholm map of index zero.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnold, V.I.: On the differential geometry of infinite-dimensional Lie groups and its application to the hydrodynamics of perfect fluids, in Vladimir I. Arnold: collected works vol. 2, Springer, New York, 2014Google Scholar
  2. 2.
    Arnold V., Khesin B.: Topological Methods in Hydrodynamics. Springer, New York (1998)MATHGoogle Scholar
  3. 3.
    Benn, J.: PhD Dissertation. University of Notre Dame, 2014Google Scholar
  4. 4.
    Benn J.: Conjugate points on the symplectomorphism group. Ann. Glob. Anal. Geom. 48, 133–147 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Benn J.: The coadjoint operator, conjugate points, and the stability of ideal fluids. Arnold Math. J. 2, 249–266 (2016)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Grossman N.: Hilbert manifolds without epiconjugate points. Proc. Am. Math. Soc. 16, 1365–1371 (1965)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ebin D., Marsden J.: Diffeomorphism groups and the motion of an incompressible fluid. Ann. Math. 92, 102–163 (1970)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Ebin D., Misiołek G., Preston S.C.: Singularities of the exponential map on the volume-preserving diffeomorphism group. Geom. Funct. anal. 16, 850–868 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1966)CrossRefMATHGoogle Scholar
  10. 10.
    Kiselev A., Šverák V.: Small scale creation for solutions of the incompressible two-dimensional Euler equation. Ann. Math. 180, 1205–1220 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Luo G., Hou T.: Toward the finite-time blowup of the 3D axisymmetric Euler equations: a numerical investigation. Multisc. Model. Simul. 12(4), 1722–1776 (2014)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Misiołek G.: Stability of ideal fluids and the geometry of the group of diffeomorphisms. Indiana Univ. Math. J. 42, 215–235 (1993)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Misiolek G.: Conjugate points in \({\mathcal{D}_\mu (\mathbb{T}^2)}\). Proc. Am. Math. Soc. 124, 977–982 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Misiołek G., Preston S.C.: Fredholm properties of Riemannian exponential maps on diffeomorphism groups. Invent. math. 179, 191–227 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Preston S.C.: On the volumorphism group, the first conjugate point is always the hardest. Commun. Math. Phys. 267, 493–513 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Preston S.C.: The WKB method for conjugate points in the volumorphism group. Indiana Univ. Math. J. 57, 3303–3327 (2008)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Preston S.C., Washabaugh P.: The geometry of axisymmetric ideal fluid flows with swirl. Arnold Math. J. 3, 175–185 (2017)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Shnirelman A.: Generalized fluid flows, their approximations and applications. Geom. Funct. anal. 4, 586–620 (1994)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Shnirelman A.: Microglobal analysis of the Euler equations. J. Math. Fluid Mech. 7, S387–S396 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Smale S.: An infinite dimensional version of Sard’s Theorem. Am. J. Math. 87, 861–865 (1965)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Wolibner W.: Un theoréme sur l’existence du mouvement plan d’un fluide parfait, homogéne, incompressible, pendant un temps infiniment long. Math. Z. 37, 698–726 (1933)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Massey University, Palmerston NorthPalmerston NorthNew Zealand
  2. 2.Department of MathematicsNotre Dame UniversityNotre DameUSA
  3. 3.Department of MathematicsBrooklyn College and CUNY Graduate CenterBrooklynUSA

Personalised recommendations