The Mabuchi geometry of low energy classes

Let $$(X,\omega )$$ be a Kähler manifold and $$\psi : \mathbb R \rightarrow R_+$$ concave weight. We show that $${\mathcal {H}}_\omega $$ admits natural metric $$d_\psi whose completion is the low energy space {E}}_\psi , introduced by Guedj–Zeriahi. As not induced Finsler metric, main difficulty to triangle inequality holds. study properties of resulting complete $$({\mathcal ,d_\psi .