  By a theorem of Greene and Wu, a noncompact connected Riemannian manifold
admits a smooth strictly subharmonic exhaustion function. Demailly provided an
elementary proof of this fact. A further simplification of Demailly's proof and
some (mostly known) applications are described. Applications include the fact
that the holomorphic line bundle associated to a nontrivial effective divisor
on a compact connected complex manifold admits a smooth Hermitian metric with
positive scalar curvature.
