manuscripta math. 78, 417 - 443 (1993) manuSCripta

mathematica

© Spnnger-Verlag 1993

ON THE SINGULAR SET OF STATIONARY HARMONIC MAPS

Fabrice BETHUEL

Let M and N be compact riemannian manifolds, and и a stationary harmonic map from M to iV^. We prove that Я'*"^(Е) = 0, where n = dim M and E is the singular set of u. This is a generalization of a result of C. Evans [7], where this is proved in the special case N is a. sphere. We also prove that, if w is a weakly harmonic map in W^*^(M,N)y then и is smooth. This extends results of F. Hélein for the case n = 2, or the case iV^ is a sphere ([9], [10]).

I Introduction

Let (M, g) and (iV, h) be two compact riemannian manifolds of dimension n and к respectively. We assume furthermore that dN = 0. We may also assume (using Nash-Moser Theorem) that N is isometrically imbedded in some euclidean space JE^. We consider the Sobolev space H^{MjN) defined by

H\M , N ) = {ue Н\М,Ш^\ u{x) e N a.e},

and for a map и in H^{MyN), the energy functional E{u) = Д^ |Vi/p.

We say that a map и in H^{MyN) is a weakly harmonic map if w is a critical point of E{u) in the following sense

Vy . € СГ(М, JR^), ^£(П(и + V)) = 0, (1.1)

where П denotes the aearest point projection onto N. It is easy to verify that (I.l) is equivalent to the following system

- A , u=^ ( « ) ( Vu , Vu ) ,

( 1 . 2 )