The Mabuchi geometry of low energy classes

Let $(X,ω)$ be a Kähler manifold and $ψ: \Bbb R \to \Bbb R_+$ be a concave weight. We show that $\mathcal H_ω$ admits a natural metric $d_ψ$ whose completion is the low energy space $\mathcal E_ψ$, introduced by Guedj-Zeriahi. As $d_ψ$ 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 $(\mathcal E_ψ,d_ψ)$.

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