On Harmonic Maps with Finite Energy

§ 1. IntroductionIn [1], R. Schoen and S. T. Yan used harmonic map to study the topology of a non-compact complete stable hypersurface in a manifold of non-negative curvature. Essentially the method is to prove first the nonexistence of some kind of nonconstant harmonic map and then to draw the relative conclusions concerning the topology. Such nonexistence theorems have been obtained in [3] and [4] in cases where the concerned manifolds are compact. The aim of this paper is to study the non-compact complete case. Our main results can be stated as follows.