The Mabuchi geometry of low energy classes

TL;DR

This paper introduces a new metric $d_ ext{psi}$ on the space of Kähler potentials, extending it to a complete low energy space, and studies its geometric properties despite it not being Finslerian.

Contribution

It constructs and analyzes a natural metric on low energy classes of Kähler potentials, extending the Mabuchi geometry to a broader setting.

Findings

The metric $d_ ext{psi}$ is well-defined and complete on $ ext{E}_ ext{psi}$.

The space $( ext{E}_ ext{psi}, d_ ext{psi})$ has specific geometric properties.

The triangle inequality holds despite the metric not being Finslerian.

Abstract

Let $(X,\omega)$ be a K\"ahler manifold and $\psi: \Bbb R \to \Bbb R_+$ be a concave weight. We show that $\mathcal H_\omega$ admits a natural metric $d_\psi$ whose completion is the low energy space $\mathcal E_\psi$, introduced by Guedj-Zeriahi. As $d_\psi$ is not induced by a Finsler metric, the main difficulty is to show that the triangle inequality holds. We study properties of the resulting complete metric space $(\mathcal E_\psi,d_\psi)$.