(Pluri)Potential Compactifications

Research output: Contribution to journalArticle

Abstract

Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.

Original languageEnglish (US)
JournalPotential Analysis
DOIs
StatePublished - Jan 1 2019

Fingerprint

Pluricomplex Green Function
Compactification
Plurisubharmonic Function
Imbedding
Potential Theory
Complex Manifolds
Compact Set
Pole
Closure
Norm
Invariant

Keywords

  • Martin boundary
  • Pluripotential theory
  • Plurisubharmonic functions

ASJC Scopus subject areas

  • Analysis

Cite this

(Pluri)Potential Compactifications. / Poletsky, Evgeny Alexander.

In: Potential Analysis, 01.01.2019.

Research output: Contribution to journalArticle

@article{3d3547ad5d4e4d41a780cbf02a253709,
title = "(Pluri)Potential Compactifications",
abstract = "Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.",
keywords = "Martin boundary, Pluripotential theory, Plurisubharmonic functions",
author = "Poletsky, {Evgeny Alexander}",
year = "2019",
month = "1",
day = "1",
doi = "10.1007/s11118-019-09766-y",
language = "English (US)",
journal = "Potential Analysis",
issn = "0926-2601",
publisher = "Springer Netherlands",

}

TY - JOUR

T1 - (Pluri)Potential Compactifications

AU - Poletsky, Evgeny Alexander

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.

AB - Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.

KW - Martin boundary

KW - Pluripotential theory

KW - Plurisubharmonic functions

UR - http://www.scopus.com/inward/record.url?scp=85060848074&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060848074&partnerID=8YFLogxK

U2 - 10.1007/s11118-019-09766-y

DO - 10.1007/s11118-019-09766-y

M3 - Article

AN - SCOPUS:85060848074

JO - Potential Analysis

JF - Potential Analysis

SN - 0926-2601

ER -