The principle of least action in the space of K\"ahler potentials

TL;DR

This paper explores the geometric structure of the space of K"ahler potentials on a compact K"ahler manifold, demonstrating that geodesics are paths of least action and establishing convexity properties that generalize previous results.

Contribution

It introduces a framework connecting geodesics and least action in the space of K"ahler potentials, extending earlier work by Calabi, Chen, and Darvas.

Findings

Geodesics of the Mabuchi connection are paths of least action.

Under certain conditions, paths of least action are geodesics.

The paper proves a convexity property of the least action in this geometric setting.

Abstract

Given a compact K\"ahler manifold, the space $\mathcal H$ of its (relative) K\"ahler potentials is an infinite dimensional Fr\'echet 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.