Non-existence of continuous convex functions on certain Riemannian manifolds

Abstract

The present note is written in response to a question of Wu. In [51 Wu proved that every complete non-compact convex hypersurface in euclidean space has infinite volume. He then asked whether this is true for all complete non-compact non-negatively curved manifold or not. The structure theorem of Cheeger and Gromoll, I-2], says that such a manifold has a continuous convex exhaustion. On the other hand, a theorem of Bishop and O'Neill, [1], says that there is no non-trivial smooth convex function on a complete manifold with finite volume. However, as was remarked by Greene and Wu [3], the passage from continuity to smoothness is highly non-trivial. The main theorem here is that there is no non-trivial continuous convex function on a complete manifold with finite volume. I would like to thank Professor Wu for his constant interest in this problem. Recently he informed us that he and Greene also proved that a complete non-compact manifold with non-negative curvature has infinite volume.