  The first three sections of this paper are a survey of the author's work on
balanced metrics and stability notions in algebraic geometry. The last section
is devoted to proving the well-known result that a geodesically convex function
on a complete Riemannian manifold admits a critical point if and only if its
asymptotic slope at infinity is positive, where we present a proof which relies
only on the Hopf--Rinow theorem and extends to locally compact complete length
metric spaces.
