  We study/construct (proper and non-proper) Morse functions on complete
Riemannian manifolds, the level hypersurfaces of which have positive mean
curvatures at all non-critical points. We show, for instance, that if a
complete Rieannin manifold admits no such (not necessarily proper) function,
then it contains a (possibly, singular) complete (possibly, compact) minimal
hypersurface of finite volume.
