Maximal subextensions of plurisubharmonic functions

Abstract

In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact Kähler manifold. We prove that a precise bound on the complex Monge-Ampère mass of the given function implies the existence of a subextension to a bigger regular subdomain or to the whole compact manifold. In some cases we show that the maximal subextension has a well defined complex Monge-Ampère measure and obtain precise estimates on this measure. Finally we give an example of a plurisubharmonic function with a well defined Monge-Ampère measure and the right bound on its Monge-Ampère mass on the unit ball in C n for which the maximal subextension to the complex projective space P n does not have a globally well defined complex Monge-Ampère measure.

Degenerate complex Monge-Ampère equations with non-Kähler forms in bounded domains

“…The set K(Ω, ω) corresponds to K(Ω, ω, φ) with φ = 0. When ω = dd c ρ, the sets F (Ω, ω) and E χ (Ω, ω) coincide with their counterparts given in [CKZ11].…”

Weighted Green functions for complex Hessian operators

“…This construction is classical in Potential Theory and has been considered also in different contexts in Pluripotential Theory (see [BT76,BT82], [CKZ11], [GLZ19], [BZ20]).…”

A viscosity approach to degenerate complex Monge-Ampère equations

“…By the subextension Theorem (see [CKZ11] for m = n), it follows that there exists w ∈ E 1 m (Ω) such that w ≤ w ′ in Ω ′ and E Ω (w) ≤ E Ω ′ (w ′ ). Since w ≤ w ′ in Ω ′ , it follows that I Ω ′ ,µ ′ (w ′ ) ≤ I Ω,µ (w), hence…”