Existence of expanders of the harmonic map flow

Abstract

We investigate the existence of weak expanding solutions of the harmonic map flow for maps with values into a smooth closed Riemannian manifold. We prove the existence of such solutions in case the target manifold is isometrically embedded as a hypersurface of some Euclidean space and the initial condition is a Lipschitz map that is homotopic to a constant.

37 pages ; Exposition of the paper re-organized ; The statement of the main Theorem has been changed to only allow target manifolds which are hypersurfaces of some Euclidean space. To appear in Annales de l'École Normale Supérieure