Variational Methods pp 309-318 | Cite as

# Harmonic Diffeomorphisms between Riemannian Manifolds

## Abstract

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 C^{1} — 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 S^{3} to S^{3}: this is a harmonic diffeomorphism but the infimum of the energy in its homotopy class is zero (see [ES]).

## Keywords

Riemannian Manifold Homotopy Class Coarea Formula Target Manifold Homeomorphismes Quasiconformes## Preview

Unable to display preview. Download preview PDF.

## References

- [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 - [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 - [C H]J. - M. CORON, F. HELEIN,
*Harmonic diffeomorphisms*,*minimizing har- monic maps and rotational symmetry*,to appear*in*Comp. Math.Google Scholar - [C G]J. - M. CORON, R. GULLIVER, p —
*harmonic maps into spheres*, preprint.Google Scholar - [E S]J. EELLS, J H SAMPSON,
*Harmonic mappings of Riemannian manifolds*, Amer. J. Math. 86 (1964), p. 109–160.Google Scholar - [G W]R. GULLIVER, B. WHITE,
*On convergence rates of Harmonic maps near points of discontinuity*,to appear in Math. Ann.Google Scholar - [Ha]
- [Hé 1]F HELEIN,
*Regularity and uniqueness of harmonic maps into an ellipsoid*,Manuscripta Math.**60 (**1988), p. 235–257.Google Scholar - [Hé 2]F. HELEIN,
*Homéomorphismes quasiconformes entre variétés Riemanniennes*,to appear in C. R. Acad. Sci. Paris.Google Scholar - [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 - [S U]J. SACKS, K. UHLENBECK,
*The existence of minimal immersions of two spheres*,Ann. Math.**113 (**1981), p. 1–24.Google Scholar