Two variational problems in K\"ahler geometry

TL;DR

This paper studies variational problems in Kähler geometry involving Monge-Ampère energy, establishing existence, uniqueness, and characterization of solutions, and applying them to construct and analyze hermitian metrics on vector bundles.

Contribution

It introduces new variational problems in Kähler geometry, proves their solutions' existence and uniqueness, and applies these to construct hermitian metrics with specific curvature properties.

Findings

Solutions to the variational problems exist and are unique.

Extremals can be used to construct hermitian metrics.

The curvature of these metrics has been investigated.

Abstract

On a K\"ahler manifold we consider the problems of maximizing/minimizing Monge--Amp\`ere energy over certain subsets of the space of K\"ahler 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.