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

Laszlo Lempert

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arXiv · 2009.09949

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

Laszlo Lempert

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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.

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Laszlo Lempert. 2020-09-21. The principle of least action in the space of K\"ahler potentials. https://arxiv.org/abs/2009.09949

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