The principle of least action in the space of Kähler potentials

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 we prove a certain convexity property of the least action. This generalizes earlier results of Calabi, Chen, and Darvas.

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Research partially supported by NSF grant DMS 1764167.

Received 11 February 2021

Received revised 15 April 2021

Accepted 29 July 2021

Published 30 November 2022