The principle of least action in the space of Kähler potentials

Abstract

Given a compact Kähler manifold, the space $\mathcal{H}$ of its (relative) Kähler potentials is an infinite dimensional Fréchet 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; under suitable conditions the converse also holds; and prove a certain convexity property of the least action. This generalizes earlier results of Calabi, Chen, and Darvas.

arXiv commentAdded to version 1: Example 5.4 of a weak geodesic with little regularity, and Sections 10, 11 to show that under suitable conditions weak geodesics are unique minimizers of the action. Some inaccuracies in version 1 corrected