Miran Černe

Analytic discs attached to a generating CR-manifold

Perturbations by analytic discs along generating CR-submanifolds of Cn are considered. In the case all partial indices of a closed pathp in a generating CR-fibration {M(ξ)}ξ∈∂D are greater or equal to −1 we can completely parametrize all small holomorphic perturbations of the pathp along the fibration {M(ξ)}ξ∈∂D. In this case we also study the geometry of perturbations by analytic discs and their relation to the conormal bundle of the fibration.

Generalized ahlfors functions

Let Σ be a bordered Riemann surface with genus g and m boundary components. Let {γ z } z∈∂Σ be a smooth family of smooth Jordan curves in C which all contain the point 0 in their interior. Let p ∈ Σ and let F be the family of all bounded holomorphic functions f on S such that f(p) > 0 and f(z) ∈ γz for almost every z ∈ ∂Σ. Then there exists a smooth up to the boundary holomorphic function f 0 ∈ F with at most 2g+m-1 zeros on S so that f 0 (z) ∈ γ z for every z ∈ ∂Σ and such that f 0 (p) ≥ f(p) for every f ∈ F. If, in addition, all the curves {γ z } z∈∂Σ are strictly convex, then f 0 is unique among all the functions from the family F.

Maximal plurisubharmonic functions and the polynomial hull of a completely circled fibration

AbstractLetX(-ϱBm×Cn be a compact set over the unit sphere ϱBm such that for eachz∈ϱBm the fiberXz={ω∈Cn;(z, ω)∈X} is the closure of a completely circled pseudoconvex domain inCn. The polynomial hull

Regularity of discs attached to a submanifold ofCn

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Boundary Value Problems for Holomorphic Functions on the Upper Half-plane

ofX is described in terms of the Perron-Bremermann function for the homogeneous defining function ofX. Moreover, for each point (z0,w0)∈Int

Embedding some bordered Riemann surfaces int the affine plane

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