On univalent harmonic maps between surfaces

Abstract

Hence the energy defines a functional on the space of Lipshitz maps between M and M'. Critical points of this functional are called harmonic maps. These maps were studied by Bochner, Morrey, Rauch, Eells and Sampson, Hartman, Uhlenbeck, Hamilton, Hildebrandt and others. The first fundamental result was due to Eells and Sampson [3] who proved that, in case M' has non-positive sectional curvature, each map from M to M' is homotopic to a harmonic map. (This result was then extended by Hamil ton [8] to the case where both M and M' are allowed to have boundary.) Later Har tman [7] was able to prove the harmonic map is unique in each homotopy class if M' has strictly negative curvature. This last result of Har tman leads one to believe that harmonic maps between compact manifolds with negative curvature must enjoy a lot of nice properties. In fact, a few years ago, B. Lawson and the second author conjectured the following statement: If f is a harmonic map between two compact Riemannian manifolds of negative curvature and if f is a homotopy equivalence, then f is a diffeomorphism. In this paper, we demonstrate that the above statement is true at least when dim M = d i m M ' = 2 . In other words, we prove that when M' has non-positive curvature and genus M = g > 1, then every degree one harmonic map from M into M' is a diffeomorphism. We also generalize this theorem to the case where both M and M' have boundary, ~M' has non-negative geodesic curvature and the harmonic map restricted to ~M is a homeomorphism from OM to t~M'. It should be noted that in case both M and M' are bounded simply connected domins in the plane, the last theorem was an old theorem and was