no. 4
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

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 .

Soit D⊂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⊂D une borne supérieure quantitative pour sup z∈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∈ℂ n /{O}, si w∈∂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 ∈∂D de type fini de façon non tangentielle, est toujours supérieur à 1∖N, où N est l’ordre d’extensibilité pseudoconvexe de ∂D en z 0 .

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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},
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     year = {2000},
     doi = {10.5802/aif.1790},
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     mrnumber = {2001k:32058},
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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

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