The Geometry of m-Hyperconvex Domains

Abstract

We study the geometry of m-regular domains within the CaffarelliNirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every mhyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.

Extension and approximation ofm-subharmonic functions

“…Then it is also locally m-hyperconvex. By Theorem 3.3 in [2], we know that Ω must be globally m-hyperconvex. Thus, there exists ψ ∈ SH m (Ω) ∩ C(Ω), ψ ≡ 0, such that ψ| ∂Ω = 0.…”

“…We shall need the following lemma. Since Ω is also m-hyperconvex there exists a smooth and strictly m-subharmonic exhaustion function ϕ for Ω (see [2]). Let M > 1 be a constant large enough so that for all z ∈ K ϕ(z) − 1 > M ψ(z) .…”

On $$\varvec{m}$$ m -Subharmonic Ordering of Measures

“…where ρ ∈ E 0,m (Ω) ∩ C ∞ (Ω) is a strictly m-subharmonic exhaustion function of Ω (see [2]). Hence, to prove u = v it is enough to show that…”

The classification of holomorphic (m, n)-subharmonic morphisms

“…Thus, we can characterize those functions f : Ω → Ω ′ such that condition (1), or (2), holds. Even though the work of Jacobi [40] is certainly an early encounter with harmonic morphisms as we know them today, the geometrical idea behind the equivalence of (2) and (3) is due to Johann Carl Friedrich Gauß [28] from 1822 (published in [29]; for an English translation see [30,31]). In modern times, the above result was used for example in analytic multivalued function theory (see e.g.…”

“…On the other hand, there has been some progress obtaining a Poisson type formula for SH k using Choquet theory and representing measures, instead of Poletsky disks (see e.g. Theorem 2.8 in [2] and the references therein).…”