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

Balanced metrics for extremal K\"ahler metrics and Fano manifolds