Plurisubharmonic envelopes and supersolutions

By Vincent Guedj, Chinh H. Lu, Ahmed Zeriahi. Journal of differential geometry, 2017

60 citations recorded by the source corpus.

We make a systematic study of (quasi-)plurisubharmonic envelopes on compact K\"ahler manifolds, as well as on domains of $\mathbb{C}^n$, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution of a given complex Monge-Amp\`ere equation. We use these ideas to solve complex Monge-Amp\`ere equations by taking lower envelopes of super-solutions.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/4f0ccbb53cb38e107bd7cdd02daaa4d3039c479e

Citations in the corpus

• Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity (2017, unknown): “ X θn Pθ[φ] = X θn φ. In our study, we will need the following preliminary result, providing an estimate for the complex Monge-Amp`ere operator of rooftop envelopes, that builds on recent progress in [GLZ17]: Lemma 3.7. Let ϕ,ψ∈ PSH(X,θ). If Pθ(ϕ,ψ) 6= −∞ then θn Pθ(ϕ,ψ) ≤ 1{Pθ(ϕ,ψ)=ϕ}θ n ϕ+ {P θ(ϕ,”
• The metric geometry of singularity types (2019, unknown): “n the singularity type [ψ]. When χ= Vθ, we will simply write P[ψ] := Pθ[ψ] := Pθ[ψ](Vθ) and call this potential the envelope of the singularity type [ψ]. It follows from [DDL2, Theorem 3.8], [Ber18], [GLZ17] that θn P[ψ] ≤ 1 {P[ψ]=0}θ n. Also, by [DDL3, Proposition 2.3 and Remark 2.5] we have that R”