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 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 be a harmonic -diffeomorphism between and . In the case where and have nonempty boundaries, we assume that the restriction of to is a diffeomorphism between and . Then we want to know if is or is not a minimizing harmonic map, i.e. if minimizes the energy functional among the maps which have the same boundary data as and which are homotopic to . In the case where and have empty boundaries, the answer is generally no because of the counterexample of the identity map from to : this is a harmonic diffeomorphism but the infimum of the energy in its homotopy class is zero [see J. Eells and J. H. Sampson, Am. J. Math. 86, 109-160 (1964; Zbl 0122.401)].