To the geometry of spaces of plurisubharmonic functions on a Kähler manifold

Abstract

Consider a compact Kähler manifold $(X,\omega )$ and the space $\mathcal{E}(\mathcal{X},\omega )=\mathcal{E}$ of $\omega$--plurisubharmonic functions of full Monge--Ampère mass on it. We introduce a quantity $\rho [u,v]$ to measure the distance between $u,v\in \mathcal{E}$; $\rho [u,v]$ is not a number but rather a decreasing function on a certain interval $(0,V)\subset \mathbb{R}$. We explore properties of $\rho [u,v]$, and using them we study Lagrangians and associated energy spaces of $\omega$--plurisubharmonic functions. Many results here generalize Darvas's findings about his metrics $d_{\chi }$.

Cited by in corpus (1)

• A complete metric topology on relative low energy spaces