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

Diederich, Klas; Herbort, Gregor

doi:10.5802/aif.1790

Diederich, Klas ; Herbort, Gregor

Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1205-1228.

Résumé
Abstract

Soit $D \subset ℂ^{n}$ un domaine pseudoconvexe qui admet une fonction plurisousharmonique d’exhaustion et Hölder continue. On note $G_{D} (., .)$ la fonction pluricomplexe de Green, pour $D$ . Dans cet article nous allons donner pour un ensemble compact $K \subset D$ une borne supérieure quantitative pour ${sup}_{z \in K} | G_{D} (z, w) |$ , à l’aide de la distance au bord de $K$ et du point $w$ . Comme application nous pouvons démontrer que, dans un domaine régulier $D$ (au sens de Diederich-Fornaess), la métrique de Bergman différentielle $B_{D} (w; X)$ tend vers l’infini, pour $X \in ℂ^{n} / {O}$ , si $w \in \partial D$ tend vers un point du bord de $D$ . De plus, nous démontrons que l’ordre de croissance de $B_{D} (W; .)$ , quand $w$ tend vers un point $z^{0} \in \partial D$ de type fini de façon non tangentielle, est toujours supérieur à $1 ∖ N$ , où $N$ est l’ordre d’extensibilité pseudoconvexe de $\partial D$ en $z^{0}$ .

Let $D \subset ℂ^{n}$ 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 ${sup}_{z \in K} | G_{D} (z, w) |$ 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 $X \in ℂ^{n} / {O}$ , 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 $1 ∖ N$ , where $N$ denotes the order of pseudoconvex extendability of $\partial D$ at $z^{0}$ .

MR Zbl

DOI : 10.5802/aif.1790

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     author = {Diederich, Klas and Herbort, Gregor},
     title = {Quantitative estimates for the {Green} function and an application to the {Bergman} metric},
     journal = {Annales de l'Institut Fourier},
     pages = {1205--1228},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {4},
     year = {2000},
     doi = {10.5802/aif.1790},
     mrnumber = {2001k:32058},
     zbl = {0960.32022},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.1790/}
}

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Diederich, Klas; Herbort, Gregor. Quantitative estimates for the Green function and an application to the Bergman metric. Annales de l'Institut Fourier, Tome 50 (2000) no. 4, pp. 1205-1228. doi : 10.5802/aif.1790. https://www.numdam.org/articles/10.5802/aif.1790/

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