$L^{1}$ metric geometry of potentials with prescribed singularities on compact Kähler manifolds

Abstract

Given (X, ω) compact Kähler manifold and ψ ∈ M + ⊂ P SH(X, ω) a model type envelope with non-zero mass, i.e. a fixed potential determing some singularities such that X (ω +dd c ψ) n > 0, we prove that the ψ−relative finite energy class E 1 (X, ω, ψ) becomes a complete metric space if endowed of a distance d which generalizes the well-known d 1 distance on the space of Kähler potentials. Later, for A ⊂ M + total ordered, we equip the set X A := ψ∈A E 1 (X, ω, ψ) of a natural distance d A which coincides with the distance d on E 1 (X, ω, ψ) for any ψ ∈ A. We show that X A , d A is a complete metric space. As a consequence, assuming ψ k ց ψ and ψ k , ψ ∈ M + , we also prove that E 1 (X, ω, ψ k ), d converges in a Gromov-Hausdorff sense to E 1 (X, ω, ψ), d and that it is possible to define a direct system E 1 (X, ω, ψ k ), P k,j in the category of metric spaces whose direct limit is dense into E 1 (X, ω, ψ), d .

Geodesics in the space of $m$-subharmonic functions with bounded energy

“…However, some unwanted properties occur. For example, take the case m = n, and for a, b < 0 define the following functions in the unit ball in C n : To avoid the mentioned complications, we will let ourselves be inspired by [56,57,59] and introduce the following metric structure. We define d :…”

Comparison of Monge-Ampère capacities

“…A. Trusiani has recently proved in [25] a comparison of Monge-Ampère φ-capacities for model potential φ using the metric geometry of relative finite energy classes introduced in [24].…”