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.

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 .

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 .

@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, Tome 50 (2000) no. 4, pp. 1205-1228. doi : 10.5802/aif.1790. https://aif.centre-mersenne.org/articles/10.5802/aif.1790/

[1] Z. Blocki, Estimates for the complex Monge-Ampère operator, Bull. Pol. Acad. Sci., 41 (1993), 151-157. | MR | Zbl

[2] Z. Blocki, P. Pflug, Hyperconvexity and Bergman completeness, Nagoya Math. J., 151 (1998), 221-225. | MR | Zbl

[3] M. Carlehed, Comparison of the pluricomplex and the classical Green functions, Michigan Math. J., 45 (1998), 399-407. | MR | Zbl

[4] M. Carlehed, U. Cegrell, F. Wikström, Jensen measures, hyperconvexity, and boundary behavior of the pluricomplex Green's function, Ann. Pol. Math., 71 (1999), 87-103. | Zbl

[5] J. P. Demailly, Mesures de Monge-Ampère et mesures pluriharmoniques, Math. Z., 194 (1987), 519-564. | MR | Zbl

[6] K. Diederich, J. E. Fornæss, Pseudoconvex domains : Bounded strictly plurisubharmonic exhaustion functions, Invent. Math., 39 (1977), 129-141. | MR | Zbl

[7] K. Diederich, J. E. Fornæss, Pseudoconvex domains with real analytic boundary, Ann. Math., 107 (1978), 371-384. | MR | Zbl

[8] K. Diederich, G. Herbort, Geometric and analytic boundary invariants on pseudoconvex domains. Comparison results, J. Geom. Analysis 3 (1993), 237-267. | MR | Zbl

[9] K. Diederich, G. Herbort, Pseudoconvex domains of semiregular type, Contributions to Complex Analysis (H. Skoda, J. M. Trépreau, ed.), Aspects of Mathematics, vol. E 26, Vieweg-Verlag, 1994, pp. 127-162. | MR | Zbl

[10] K. Diederich, T. Ohsawa, An estimate on the Bergman distance on pseudoconvex domains, Ann. of Math., 141 (1995), 181-190. | MR | Zbl

[11] I. Graham, Boundary behavior of the Caratheodory and Kobayashi metrics on strongly pseudoconvex domains in ℂn with smooth boundary, Trans. Amer. Math. Soc., 207 (1975), 219-240. | MR | Zbl

[12] G. Herbort, The Bergman metric on hyperconvex domains, Math. Zeit., 232 (1999), 183-196. | MR | Zbl

[13] G. Herbort, On the pluricomplex Green function on pseudoconvex domains with a smooth boundary, Internat. J. of Math. (To appear). | Zbl

[14] L. Hörmander, An introduction to complex analysis in several variables, North Holland, Amsterdam, 2nd ed. ed., 1973. | Zbl

[15] M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances, Bull. Soc. Math. France, 113 (1985), 231-240. | Numdam | MR | Zbl

Cité par Sources :