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,...

We give several definitions of the pluricomplex Green function and show their equivalence.

Using a generalization of [Pol] we present a description of complex geodesics in arbitrary complex ellipsoids.

We give a simplified proof of J. P. Rosay's result on plurisubharmonicity of the envelope of the Poisson functional [10].

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.

We show that the symmetrized bidisc 𝔾₂ = {(λ₁+λ₂,λ₁λ₂):|λ₁|,|λ₂| < 1} ⊂ ℂ² cannot be exhausted by domains biholomorphic to convex domains.

We characterize proper holomorphic self-mappings 𝔾₂ → 𝔾₂ for the symmetrized bidisc 𝔾₂ = {(λ₁+λ₂,λ₁λ₂): |λ₁|,|λ₂| < 1} ⊂ ℂ².

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.

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.

Let $A$ be a closed polar subset of a domain $D$ in $\u2102$. We give a complete
description of the pluripolar hull ${\Gamma}_{D\times \u2102}^{*}$ of the graph $\Gamma $ of a
holomorphic function defined on $D\setminus A$. To achieve this, we prove for
pluriharmonic measure certain semi-continuity properties and a localization principle.

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.

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.

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.

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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