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Harmonic diffeomorphisms between Riemannian manifolds. (English) Zbl 0738.58015
Variational methods, Proc. Conf., Paris/Fr. 1988, Prog. Nonlinear Differ. Equ. Appl. 4, 309-318 (1990).
[For the entire collection see Zbl 0713.00009.] In this lecture I will speak principally about a joint work with {\it J.- M. Coron} [Compos. Math. 69, No. 2, 175-228 (1989; Zbl 0686.58012)] 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 $C\sp 1$-diffeomorphism between $M$ and $N$. In the case where $M$ and $N$ have nonempty boundaries, we assume that the restriction of $u$ to $\partial M$ is a diffeomorphism between $\partial M$ and $\partial N$. 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 $S\sp 3$ to $S\sp 3$: this is a harmonic diffeomorphism but the infimum of the energy in its homotopy class is zero [see {\it J. Eells} and {\it J. H. Sampson}, Am. J. Math. 86, 109-160 (1964; Zbl 0122.401)].

58E20Harmonic maps between infinite-dimensional spaces
53C20Global Riemannian geometry, including pinching
35J50Systems of elliptic equations, variational methods