Quantitative estimates for the Green function and an application to the Bergman metric

Klas Diederich; Gregor Herbort

Annales de l'institut Fourier (2000)

  • Volume: 50, Issue: 4, page 1205-1228
  • ISSN: 0373-0956

Abstract

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Let D ⊂ ℂ n be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by G D ( . , . ) . In this article we give for a compact subset K ⊂ D a quantitative upper bound for the supremum sup z ∈ K | G D ( z , w ) | in terms of the boundary distance of K and w . This enables us to prove that, on a smooth bounded regular domain D (in the sense of Diederich-Fornaess), the Bergman differential metric B D ( w ; X ) tends to infinity, for X ∈ ℂ n / { O } , when w ∈ D tends to a boundary point. Furthermore, we prove that the order of growth of B D ( W ; . ) under nontangential approach of w ∈ D to a point z 0 ∈ ∂ D of finite type, can be estimated from below by 1 ∖ N , where N denotes the order of pseudoconvex extendability of ∂ D at z 0 .

How to cite

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Diederich, Klas, and Herbort, Gregor. "Quantitative estimates for the Green function and an application to the Bergman metric." Annales de l'institut Fourier 50.4 (2000): 1205-1228. <http://eudml.org/doc/75454>.

@article{Diederich2000,
abstract = {Let $D\subset \{\Bbb C\}^n$ be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by $G_D(.,.)$. In this article we give for a compact subset $K\subset D$ a quantitative upper bound for the supremum $\sup _\{z\in K\}\vert G_D(z,w)\vert $ in terms of the boundary distance of $K$ and $w$. This enables us to prove that, on a smooth bounded regular domain $D$ (in the sense of Diederich-Fornaess), the Bergman differential metric $B_D(w;X)$ tends to infinity, for $X\in \{\Bbb C\}^n/\lbrace O\rbrace $, when $w\in D$ tends to a boundary point. Furthermore, we prove that the order of growth of $B_D(W;.)$ under nontangential approach of $w\in D$ to a point $z^0\in \partial D$ of finite type, can be estimated from below by $1\backslash N$, where $N$ denotes the order of pseudoconvex extendability of $\partial D$ at $z^0$.},
author = {Diederich, Klas, Herbort, Gregor},
journal = {Annales de l'institut Fourier},
keywords = {pluricomplex Green function; Monge-Ampère equation; order of pseudoconvex extendability; Bergman metric},
language = {eng},
number = {4},
pages = {1205-1228},
publisher = {Association des Annales de l'Institut Fourier},
title = {Quantitative estimates for the Green function and an application to the Bergman metric},
url = {http://eudml.org/doc/75454},
volume = {50},
year = {2000},
}

TY - JOUR
AU - Diederich, Klas
AU - Herbort, Gregor
TI - Quantitative estimates for the Green function and an application to the Bergman metric
JO - Annales de l'institut Fourier
PY - 2000
PB - Association des Annales de l'Institut Fourier
VL - 50
IS - 4
SP - 1205
EP - 1228
AB - Let $D\subset {\Bbb C}^n$ be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by $G_D(.,.)$. In this article we give for a compact subset $K\subset D$ a quantitative upper bound for the supremum $\sup _{z\in K}\vert G_D(z,w)\vert $ in terms of the boundary distance of $K$ and $w$. This enables us to prove that, on a smooth bounded regular domain $D$ (in the sense of Diederich-Fornaess), the Bergman differential metric $B_D(w;X)$ tends to infinity, for $X\in {\Bbb C}^n/\lbrace O\rbrace $, when $w\in D$ tends to a boundary point. Furthermore, we prove that the order of growth of $B_D(W;.)$ under nontangential approach of $w\in D$ to a point $z^0\in \partial D$ of finite type, can be estimated from below by $1\backslash N$, where $N$ denotes the order of pseudoconvex extendability of $\partial D$ at $z^0$.
LA - eng
KW - pluricomplex Green function; Monge-Ampère equation; order of pseudoconvex extendability; Bergman metric
UR - http://eudml.org/doc/75454
ER -

References

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  10. [10] K. DIEDERICH, T. OHSAWA, An estimate on the Bergman distance on pseudoconvex domains, Ann. of Math., 141 (1995), 181-190. Zbl0828.32002MR95j:32039
  11. [11] I. GRAHAM, Boundary behavior of the Caratheodory and Kobayashi metrics on strongly pseudoconvex domains in ℂn with smooth boundary, Trans. Amer. Math. Soc., 207 (1975), 219-240. Zbl0305.32011MR51 #8468
  12. [12] G. HERBORT, The Bergman metric on hyperconvex domains, Math. Zeit., 232 (1999), 183-196. Zbl0933.32048MR2000i:32020
  13. [13] G. HERBORT, On the pluricomplex Green function on pseudoconvex domains with a smooth boundary, Internat. J. of Math. (To appear). Zbl1110.32307
  14. [14] L. HÖRMANDER, An introduction to complex analysis in several variables, North Holland, Amsterdam, 2nd ed. ed., 1973. Zbl0271.32001
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