The Mabuchi geometry of low energy classes

Abstract

Let $(X,\omega )$ be a Kähler manifold and $\psi :\mathbb{R}\to {\mathbb{R}}_{+}$ 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 $({\mathcal{E}}_{\psi },d_{\psi })$.

arXiv commentv2. updated references v.3 final version