Convex Functions and Harmonic Maps

A subset D of a riemannian manifold Y is said to be convex supporting if every compact subset of D has a Y-open neighborhood which supports a strictly convex function. The image of a harmonic map f from a compact manifold X to Y cannot be contained in any convex supporting subset of Y unless f is constant. Also, if Y has a convex supporting covering space and <!-- MATH ${\pi _1}(X)$ --> is finite then every harmonic map from X to Y is necessarily constant. Examples of convex supporting domains and manifolds are given.