Harmonic Diffeomorphisms between Riemannian Manifolds

  • Frederic Helein
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)


In this lecture I will speak principally about a joint work with J.-M. Coron[C H] in which we studied the following problem : let M and N be two Riemannian manifolds with or without boundary, which are diffeomorphic, and let u be a harmonic C1 — diffeomorphism between M and N. In the case where M and N have nonempty boundaries, we assume that the restriction of u to ∂M is a diffeomorphism between ∂M and ∂N. Then we want to know if u is or is not a minimizing harmonic map, i.e. if u minimizes the energy functional among the maps which have the same boundary data as u and which are homotopic to u. In the case where M and N have empty boundaries, the answer is generally no because of the counterexample of the identity map from S3 to S3: this is a harmonic diffeomorphism but the infimum of the energy in its homotopy class is zero (see [ES]).


Riemannian Manifold Homotopy Class Coarea Formula Target Manifold Homeomorphismes Quasiconformes 
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  1. [A]
    S.I. AL’BER, On n-dimensional Problems in the Calculus of Variations in the Large, Soviet. Math. Dokl. 5 (1964), p. 700–704.Google Scholar
  2. [B]
    A. BALDES, Stability and uniqueness properties of the equator map from a ball into an ellipsoid, Math. Z. 185 (1984), p. 505 - 516.Google Scholar
  3. [C H]
    J. - M. CORON, F. HELEIN, Harmonic diffeomorphisms, minimizing har- monic maps and rotational symmetry,to appear in Comp. Math.Google Scholar
  4. [C G]
    J. - M. CORON, R. GULLIVER, p — harmonic maps into spheres, preprint.Google Scholar
  5. [E S]
    J. EELLS, J H SAMPSON, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), p. 109–160.Google Scholar
  6. [G W]
    R. GULLIVER, B. WHITE, On convergence rates of Harmonic maps near points of discontinuity,to appear in Math. Ann.Google Scholar
  7. [Ha]
    P. HARTMAN, On homotopic harmonic maps,Canad. J. Math. 19 (1967), p. 547-570.Google Scholar
  8. [Hé 1]
    F HELEIN, Regularity and uniqueness of harmonic maps into an ellipsoid,Manuscripta Math. 60 (1988), p. 235–257.Google Scholar
  9. [Hé 2]
    F. HELEIN, Homéomorphismes quasiconformes entre variétés Riemanniennes,to appear in C. R. Acad. Sci. Paris.Google Scholar
  10. [J K]
    W. JAGER, H. KAUL. Rotationally symmetric harmonic map from a ball into a sphere and the regularity problem for weak solutions of elliptic systems,J. Reine Angew. Math. 343 (1983), p. 146–161.Google Scholar
  11. [S U]
    J. SACKS, K. UHLENBECK, The existence of minimal immersions of two spheres,Ann. Math. 113 (1981), p. 1–24.Google Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Frederic Helein
    • 1
  1. 1.Centre de MathématiquesEcole PolytechniquePalaiseau CedexFrance

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