Abstract
Using pluricomplex Green functions we introduce a compactification of a complex manifold M invariant with respect to biholomorphisms similar to the Martin compactification in the potential theory. For this we show the existence of a norming volume form V on M such that all negative plurisubharmonic functions on M are in L1(M, V ). Moreover, the set of such functions with the norm not exceeding 1 is compact. Identifying a point w ∈ M with the normalized pluricomplex Green function with pole at w we get an imbedding of M into a compact set and the closure of M in this set is the pluripotential compactification.
Similar content being viewed by others
References
Armitage, D.H., Gardiner, S.J.: Classical potential theory. Springer, Berlin (2001)
Bracci, F., Patrizio, G., Trapani, S.: The pluricomplex Poisson kernel for strongly convex domains. Trans. Amer. Math. Soc. 361, 979–1005 (2009)
Chang, C.-H., Hu, M.C., Lee, H.-P.: Extremal analytic discs with prescribed boundary data. Trans. Amer. Math. Soc. 310, 355–369 (1988)
Demailly, J.-P.: Mesures de Monge-Ampére et mesures pluriharmoniques. Math. Z. 194, 519–564 (1987)
Helms, L.L.: Introduction to Potential Theory. Wiley-Interscience, New York (1969)
Hörmander, L.: The Analysis of Linear Partial Differential Operators, I, Distribution Theory and Fourier Analysis. Springer, Berlin (1983)
Klimek, M.: Extremal plurisubharmonic functions and invariant pseudodistances. Bull. Soc. Math. France 113, 123–142 (1985)
Klimek, M.: Pluripotential theory. Clarendon Press, Oxford (1991)
Keldysch, M.V., Lavrentiev, M.A.: Sur une évalution pour la fonction de Green. Dokl. Acad. Nauk USSR 24, 102–103 (1939)
Lelong, P.: Discontinuité et annulation de l’opérateur de Monge-Ampére complexe, P. Lelong-P. Dolbeault-H. Skoda analysis seminar, 1981/1983, Lecture Notes in Math, vol. 1028, pp 219—224. Springer, Berlin (1983)
Lempert, L.: La métrique de Kobayashi et la représentation des domaines sur la boule. Bull. Soc. Math. France 109, 427–474 (1981)
Martin, R.S.: Minimal positive harmonic functions. Trans. Amer. Math. Soc. 49, 137–172 (1941)
Poletsky, E.A., Shabat, B.V.: Invariant metrics, Current problems in mathematics. Fund. Dir. 9, 73–125 (1986). Itogi Nauki i Tekhniki. Akad. Nauk SSSR
Privalov, I.I., Kuznetsov, P.K.: Boundary problems and classes of harmonic and subharmonic functions in arbitrary domains. Mat. Sb. 6, 345–375 (1939)
Solomentsev, E.D.: Boundary values of subharmonic functions. Czech. Math. J. 8, 520–534 (1958)
Vladimirov, V.S.: Equations of Mathematical Physics. Moscow, Nauka (1967)
Widman, K.-O.: Inequalities for the Green function and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand. 21, 17–37 (1967)
Zhao, Z.X.: Green function for Schrödinger operator and conditioned Feynman-Kac gauge. J. Math. Anal. Appl. 116, 309–334 (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The author was partially supported by a grant from Simons Foundation.
Rights and permissions
About this article
Cite this article
Poletsky, E.A. (Pluri)Potential Compactifications. Potential Anal 53, 231–245 (2020). https://doi.org/10.1007/s11118-019-09766-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-019-09766-y