Two variational problems in Kähler geometry

On a Kähler manifold we consider the problems of maximizing/minimizing Monge–Ampère energy over certain subsets of the space of Kähler potentials. Under suitable assumptions we prove that solutions to these variational problems exist, are unique, and have a simple characterization. We then use the extremals to construct hermitian metrics on holomorphic vector bundles, and investigate their curvature.