The principle of least action in the space of Kähler potentials

Abstract

Given a compact Kahler manifold, the space $\mathcal H$ of its (relative) Kahler potentials is an infinite dimensional Frechet manifold, on which Mabuchi and Semmes have introduced a natural connection $\nabla$. We study certain Lagrangians on $T\mathcal H$, in particular Finsler metrics, that are parallel with respect to the connection. We show that geodesics of $\nabla$ are paths of least action, and prove a certain convexity property of the least action. This generalizes earlier results of Calabi, Chen, and Darvas.