Invariant pluricomplex Green functions

Klimek, Maciej

Banach Center Publications, Tome 31 (1995), p. 207-226 / Harvested from The Polish Digital Mathematics Library

Résumé

The purpose of this paper is to present a concise survey of the main properties of biholomorphically invariant pluricomplex Green functions and to describe a number of new examples of such functions. A concept of pluricomplex geodesics is also discussed.

Détail

Publié le : 1995-01-01

Zbl 0844.31004

EUDML-ID : urn:eudml:doc:262801

Bibliographie

[000] [AP] M. Abate and G. Patrizio, Uniqueness of complex geodesics and characterization of circular domains, Manuscripta Math. 74 (1992), 277-297. | Zbl 0758.53039

[001] [A] A. Aytuna, On Stein manifolds M for which O(M) is isomorphic to $O(Δ^n)$, Manuscripta Math., 62 (1988), 297-315. | Zbl 0662.32014

[002] [A1] K. Azukawa, Two intrinsic pseudo-metrics with pseudoconvex indicatrices and starlike circular domains, J. Math. Soc. Japan 38 (1986), 627-647. | Zbl 0607.32015

[003] [A2] K. Azukawa, The invariant pseudo-metric related to negative plurisubharmonic functions, Kodai Math. J. 10 (1987), 83-92. | Zbl 0618.32020

[004] [BL] T. Bagby and N. Levenberg, Bernstein theorems for harmonic functions, preprint, Indiana University, Bloomington, 1992.

[005] [B] E. Bedford, Survey of pluri-potential theory, preprint, 1990.

[006] [BD] E. Bedford and J.-P. Demailly, Two counterexamples concerning the pluri-complex Green function in $C^n$, Indiana Univ. Math. J. 37 (1988), 865-867. | Zbl 0681.32014

[007] [BT1] E. Bedford and B. A. Taylor, The Dirichlet problem for a complex Monge-Ampère equation, Invent. Math. 37 (1976), 1-44. | Zbl 0315.31007

[008] [BT2] E. Bedford and B. A. Taylor, A new capacity for plurisubharmonic functions, Acta Math. 149 (1982), 1-40. | Zbl 0547.32012

[009] [BT3] E. Bedford and B. A. Taylor, Plurisubharmonic functions with logarithmic singularities, Ann. Inst. Fourier (Grenoble) 38 (1988), 133-171. | Zbl 0626.32022

[010] [BT4] E. Bedford and B. A. Taylor, Uniqueness for the complex Monge-Ampère equation for functions of logarithmic growth, Indiana Univ. Math. J. 38 (1989), 455-469. | Zbl 0677.32002

[011] [BŁ] Z. Błocki, Estimates for the complex Monge-Ampère operator, preprint, Jagiellonian University, Cracow, 1992.

[012] [BR] F. T. Brawn, The Green and Poisson kernels for the strip $ℝ^n ×]0,1[$, J. London Math. Soc. 2 (1970), 439-454. | Zbl 0197.08502

[013] [C1] U. Cegrell, Capacities in Complex Analysis, Aspects of Mathematics, Vieweg, Wiesbaden, 1988. | Zbl 0655.32001

[014] [C2] U. Cegrell, The symmetric pluricomplex Green function, this volume, 135-141. | Zbl 0831.31008

[015] [CLN] S. S. Chern, H. Levine and L. Nirenberg, Intrinsic norms on a complex manifold, in: Global analysis, papers in honour of K. Kodaira, University of Tokyo Press, 1969, 119-139.

[016] [D1] J. -P. Demailly, Mesures de Monge-Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. France 19 (1985), 1-125.

[017] [D2] J. -P. Demailly, Mesures de Monge-Ampère et mesures pluri-sousharmoniques, Math. Z. 194 (1987), 519-564. | Zbl 0595.32006

[018] [D] S. Dineen, The Schwarz Lemma, Oxford Math. Monographs, Clarendon Press, Oxford, 1989. | Zbl 0708.46046

[019] [DG] S. Dineen and F. Gaughran, Maximal plurisubharmonic functions on domains in Banach spaces, preprint, University College Dublin, 1992. | Zbl 1080.46517

[020] [DT] S. Dineen and R. M. Timoney, Complex geodesics on complex domains, in: Progress in Functional Analysis, K. D. Bierstedt, J. Bonet, J. Horváth and M. Maestre (eds.), Elsevier, 1992, 333-365. | Zbl 0785.46044

[021] [H1] M. Hervé, Lindelöf's principle in infinite dimensions, in: Proceedings on Infinite Dimensional Holomorphy, T. L. Hayden and T. J. Suffridge (eds.), Lecture Notes in Math. 364, Springer, Berlin, 1974, 41-57.

[022] [H2] M. Hervé, Analyticity in Infinite Dimensional Spaces, de Gruyter Stud. Math. 10, Walter de Gruyter, Berlin, 1989.

[023] [JP1] M. Jarnicki and P. Pflug, Invariant pseudodistances and pseudometrics - completeness and product property, Ann. Polon. Math. 60 (1991), 169-189. | Zbl 0756.32016

[024] [JP2] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter & Co, 1993. | Zbl 0789.32001

[025] [K] C. O. Kiselman, Sur la définition de l'opérateur de Monge-Ampère complexe, in: Analyse complexe, proceedings, Toulouse 1983, E. Amar, R. Gay, and Nguyen Thanh Van (eds.), Lecture Notes in Math. 1094, Springer, Berlin, 1983, 139-150.

[026] [K1] M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances, Bull. Soc. Math. France 113 (1985), 231-240. | Zbl 0584.32037

[027] [K2] M. Klimek, Infinitesimal pseudometrics and the Schwarz lemma, Proc. Amer. Math. Soc. 105 (1989), 134-140. | Zbl 0681.32018

[028] [K3] M. Klimek, Pluripotential Theory, London Math. Soc. Monographs (New Series), Clarendon Press, Oxford, 1991.

[029] [KR] S. L. Krushkal, The Green function of Teichmüller spaces with applications, Bull. Amer. Math. Soc. 27 (1992), 143-147.

[030] [L] P. Lelong, Fonction de Green pluricomplexe pour les espaces de Banach, J. Math. Pures Appl. 68 (1989), 319-347. | Zbl 0633.32019

[031] [L1] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474.

[032] [L2] L. Lempert, Holomorphic retracts and intrinsic metrics in convex domains, Anal. Math. 8 (1982), 257-261. | Zbl 0509.32015

[033] [L3] L. Lempert, Solving the degenerate Monge-Ampère equation with one concentrated singularity, Math. Ann. 263 (1983), 515-532. | Zbl 0531.35020

[034] [M] S. Momm, Boundary behaviour of extremal plurisubharmonic functions, preprint, University of Düsseldorf, 1992.

[035] [PS] E. A. Poletskiĭ and B. V. Shabat, Invariant metrics, in: Encyclopaedia of Mathematical Sciences 9, Several Complex Variables III, G. M. Khenkin (ed.), trans. J. Peetre, Springer, Berlin, 1989, 63-111.

[036] [RT] J. Rauch and B. A. Taylor, The Dirichlet problem for the multidimensional Monge-Ampère equation, Rocky Mountain J. Math. 7 (1977), 345-364. | Zbl 0367.35025

[037] [S] A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Math. Surveys 36 (4) (1981), 61-119. | Zbl 0494.31005

[038] [SE] S. Semmes, A generalization of Riemann mappings and geometric structures on a space of domains in $C^n$, Mem. Amer. Math. Soc. 472 (1992).

[039] [SI] N. Sibony, Quelques problèmes de prolongement de courants en analyse complexe, Duke Math. J. 52 (1985), 157-97. | Zbl 0578.32023

[040] [V1] E. Vesentini, Variations on a theme of Carathéodory, Ann. Scuola Norm. Sup. Pisa 4 (1979), 39-68. | Zbl 0413.46039

[041] [V2] E. Vesentini, Complex geodesics, Compositio Math. 44 (1981), 375-394. | Zbl 0488.30015

[042] [V3] E. Vesentini, Complex geodesics and holomorphic maps, Symposia Math. 26 (1982), 211-230.

[043] [Z] A. Zeriahi, Fonction de Green pluricomplexe à pôle à l'infini sur un espace de Stein parabolique et applications, Math. Scand. 69 (1991), 89-126.