Let be a bounded pseudoconvex domain that admits a Hölder continuous plurisubharmonic exhaustion function. Let its pluricomplex Green function be denoted by . In this article we give for a compact subset a quantitative upper bound for the supremum in terms of the boundary distance of and . This enables us to prove that, on a smooth bounded regular domain (in the sense of Diederich-Fornaess), the Bergman differential metric tends to infinity, for , when tends to a boundary point. Furthermore, we prove that the order of growth of under nontangential approach of to a point of finite type, can be estimated from below by , where denotes the order of pseudoconvex extendability of at .
Soit un domaine pseudoconvexe qui admet une fonction plurisousharmonique d’exhaustion et Hölder continue. On note la fonction pluricomplexe de Green, pour . Dans cet article nous allons donner pour un ensemble compact une borne supérieure quantitative pour , à l’aide de la distance au bord de et du point . Comme application nous pouvons démontrer que, dans un domaine régulier (au sens de Diederich-Fornaess), la métrique de Bergman différentielle tend vers l’infini, pour , si tend vers un point du bord de . De plus, nous démontrons que l’ordre de croissance de , quand tend vers un point de type fini de façon non tangentielle, est toujours supérieur à , où est l’ordre d’extensibilité pseudoconvexe de en .
@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}, pages = {1205--1228}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {50}, number = {4}, year = {2000}, doi = {10.5802/aif.1790}, zbl = {0960.32022}, mrnumber = {2001k:32058}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1790/} }
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 SP - 1205 EP - 1228 VL - 50 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1790/ DO - 10.5802/aif.1790 LA - en ID - AIF_2000__50_4_1205_0 ER -
%0 Journal Article %A Diederich, Klas %A Herbort, Gregor %T Quantitative estimates for the Green function and an application to the Bergman metric %J Annales de l'Institut Fourier %D 2000 %P 1205-1228 %V 50 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1790/ %R 10.5802/aif.1790 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.1790/
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