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 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]).