Mathematical Research Letters

Volume 29 (2022)

Number 3

The principle of least action in the space of Kähler potentials

Pages: 785 – 834



László Lempert (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)


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.

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