On interior regularity and Liouville's theorem for harmonic mappings

Abstract

It is well known that the weakly harmonic mapping U∶M→N (M,N: Riemannian manifolds) is regular if the image U(M) is contained in some sufficiently small ball and for this case Liouville's theorem is valid. In this paper we show that the smallness condition for U(M) can be released if U minimizes the energy functional and the sectional curvatures of the target manifold N are bounded by some suitable function of the distance from some fixed point of N.