Balanced metrics for extremal Kähler metrics and Fano manifolds

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.