Quantitative estimates for the Green function and an application to the Bergman metric
Annales de l'Institut Fourier, Volume 50 (2000) no. 4, p. 1205-1228

Let D 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 KD a quantitative upper bound for the supremum sup zK |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 n /{O}, when wD tends to a boundary point. Furthermore, we prove that the order of growth of B D (W;.) under nontangential approach of wD to a point z 0 D of finite type, can be estimated from below by 1N, where N denotes the order of pseudoconvex extendability of D at z 0 .

Soit D 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 KD une borne supérieure quantitative pour sup zK |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 n /{O}, si wD 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 D de type fini de façon non tangentielle, est toujours supérieur à 1N, où N est l’ordre d’extensibilité pseudoconvexe de D en z 0 .

@article{AIF_2000__50_4_1205_0,
     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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {4},
     year = {2000},
     pages = {1205-1228},
     doi = {10.5802/aif.1790},
     zbl = {0960.32022},
     mrnumber = {2001k:32058},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_4_1205_0}
}
Diederich, Klas; Herbort, Gregor. Quantitative estimates for the Green function and an application to the Bergman metric. Annales de l'Institut Fourier, Volume 50 (2000) no. 4, pp. 1205-1228. doi : 10.5802/aif.1790. http://www.numdam.org/item/AIF_2000__50_4_1205_0/

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