## Analytic discs method in complex analysis

### Volume 402 / 2002

#### Abstract

\def\CC{{\sym C}}% Our work is divided into six chapters. In Chapter I we introduce necessary notions and present most important facts. We also present our main results. Chapter I covers the following topics: \vskip 4pt $\bullet$ Extremal plurisubharmonic functions: the relative extremal function and the pluricomplex Green function; $\bullet$ The analytic discs method of E.~Poletsky: disc functionals, envelope of a disc functional, examples of disc functionals; $\bullet$ The Poisson functional: We present properties of the most important functional, including the main result of the paper, plurisubharmonicity of the envelope of the Poisson functional on a class of complex manifolds. We also prove the product property of the relative extremal function; $\bullet$ The Riesz functional: We state some properties of the Riesz functional which follow from the properties of the Poisson functional and the Poisson–Jensen formula. Since these results are contained in other papers, we do not give the proofs. $\bullet$ The Green and Lelong functionals: We concentrate mainly on the product property of the Green functional. \vskip 4pt Chapter II is devoted to the general properties of disc functionals (Section~2.1, Propositions~2.1–2.5) and properties of analytic discs in complex manifolds (Section 2.3). In Section 2.2 we study a class of complex manifolds which is important in Poletsky's theory. In Chapter III we give the main results of the paper. We show that the envelope of the Poisson functional on any complex manifold is upper semicontinuous (Theorem~3.5). Section 3.2 contains the most important (and most difficult) result of the paper. In Theorem~3.10 we show the plurisubharmonicity of the Poisson functional on a class of complex manifolds. Section 3.3 contains properties of the Poisson functional on Liouville manifolds. Using Poletsky's theory, we give a characterization of Liouville manifolds in terms of analytic discs (Theorem 3.21). Product properties of the Poisson and Green functionals are presented in Chapter IV (Theorems 4.1 and 4.9). In Chapter V we give applications of the results obtained. In Section~5.1 we state some properties of the relative extremal function. In Section 5.2, using the product property of the relative extremal function for open sets (Theorem 5.3.) we show the product property of the plurisubharmonic measure in bounded domains in $\CC^n$ (Theorem 5.6). Section~5.3 is devoted to the pluricomplex Green function. We obtain the product property of the pluricomplex Green function as a corollary of the product property of the relative extremal function (Theorem 5.8). In Section 5.4 we give simple results related to the polynomial hulls of compact sets in $\CC^n$ (Theorem 5.10). Chapter VI contains remarks related to Poletsky's theory. We concentrate mainly on holomorphically invariant pseudodistances (Section 6.4). Most of the prerequisites that we use may be found in the following books: \cite{Gun-Ros}, \cite{JP1}, \cite{Kli}, \cite{Krantz}. Some of the results contained in this work may be found in the following papers: \cite{Ed1}, \cite{Ed2}, \cite{Ed3}, \cite{Ed4}, \cite{Ed5}. This research was partly supported by the Foundation for Polish Science (FNP). The author thanks Professors Marek Jarnicki, Peter Pflug and Włodzimierz Zwonek for their remarks and for stimulating discussions.