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

Diederich, Klas; Herbort, Gregor

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

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

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