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Analytic discs method in complex analysis

100%
Edigarian A.
Instytut Matematyczny Polskiej Akademii Nauk
EN
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: • Extremal plurisubharmonic functions: the relative extremal function and the pluricomplex Green function; • The analytic discs method of E. Poletsky: disc functionals, envelope of a disc functional, examples of disc functionals; • 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; • 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. • The Green and Lelong functionals: We concentrate mainly on the product property of the Green functional. 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 ℂⁿ (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 ℂⁿ (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.
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A note on Rosay's paper

95%
Edigarian A.
Annales Polonici Mathematici
|
2003
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tom 80
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nr 1
125-132
EN
We give a simplified proof of J. P. Rosay's result on plurisubharmonicity of the envelope of the Poisson functional [10].
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A note on Costara's paper

95%
Edigarian A.
Annales Polonici Mathematici
|
2004
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tom 83
|
nr 2
189-191
EN
We show that the symmetrized bidisc 𝔾₂ = {(λ₁+λ₂,λ₁λ₂):|λ₁|,|λ₂| < 1} ⊂ ℂ² cannot be exhausted by domains biholomorphic to convex domains.
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On extremal mappings in complex ellipsoids

95%
Edigarian A.
Annales Polonici Mathematici
|
1995
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tom 62
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nr 1
83-96
EN
Using a generalization of [Pol] we present a description of complex geodesics in arbitrary complex ellipsoids.
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On definitions of the pluricomplex Green function

95%
Edigarian A.
Annales Polonici Mathematici
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1997
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tom 67
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nr 3
233-246
EN
We give several definitions of the pluricomplex Green function and show their equivalence.
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Proper holomorphic self-mappings of the symmetrized bidisc

95%
Edigarian A.
Annales Polonici Mathematici
|
2004
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tom 84
|
nr 2
181-184
EN
We characterize proper holomorphic self-mappings 𝔾₂ → 𝔾₂ for the symmetrized bidisc 𝔾₂ = {(λ₁+λ₂,λ₁λ₂): |λ₁|,|λ₂| < 1} ⊂ ℂ².
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The Pluripolar Hull and the Fine Topology

95%
Edigarian A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2005
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tom 53
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nr 3
285-290
EN
We show that the projections of the pluripolar hull of the graph of an analytic function in a subdomain of the complex plane are open in the fine topology.
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Peak points for domains in ℂⁿ

95%
Edigarian A.
Annales Polonici Mathematici
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2015
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tom 114
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nr 1
1-12
EN
We give a necessary and sufficient condition for the existence of a weak peak function by using Jensen type measures. We also show the existence of a weak peak function for a class of Reinhardt domains.
9
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Almost Properness of Extremal Mappings

60%
Edigarian A., Kliś P.
Bulletin of the Polish Academy of Sciences. Mathematics
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2009
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tom 57
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nr 2
129-133
EN
We give a simple proof of almost properness of any extremal mapping in the sense of Lempert function or in the sense of Kobayashi-Royden pseudometric.
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An elementary proof of the Dalang-Morton-Willinger theorem

60%
Edigarian A., Rygiel A.
Applicationes Mathematicae
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2007
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tom 34
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nr 4
383-388
EN
L. C. G. Rogers has given an elementary proof of the fundamental theorem of asset pricing in the case of finite discrete time, due originally to Dalang, Morton and Willinger. The purpose of this paper is to give an even simpler proof of this important theorem without using the existence of regular conditional distribution, in contrast to Rogers' proof.
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On the product property for the transfinite diameter

49%
Błocki Z., Edigarian A., Siciak J.
Annales Polonici Mathematici
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2011
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tom 101
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nr 3
209-214
EN
We give a pluripotential-theoretic proof of the product property for the transfinite diameter originally shown by Bloom and Calvi. The main tool is the Rumely formula expressing the transfinite diameter in terms of the global extremal function.
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Bounded holomorphic functions with multiple sheeted pluripolar hulls

49%
Edigarian A., Siciak J., Zwonek W.
Studia Mathematica
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2006
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tom 175
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nr 3
233-247
EN
We describe compact subsets K of ∂𝔻 and ℝ admitting holomorphic functions f with the domains of existence equal to ℂ∖K and such that the pluripolar hulls of their graphs are infinitely sheeted. The paper is motivated by a recent paper of Poletsky and Wiegerinck.
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The image of a finely holomorphic map is pluripolar

49%
Edigarian A., El Marzguioui S., Wiegerinck J.
Annales Polonici Mathematici
|
2010
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tom 97
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nr 2
137-149
EN
We prove that the image of a finely holomorphic map on a fine domain in ℂ is a pluripolar subset of ℂⁿ. We also discuss the relationship between pluripolar hulls and finely holomorphic functions.
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