2021/12/11 by S. Ivashkovich, Ivashkovich, Sergey
Mathematics · #32D15 #Complex Variables (math.CV) #FOS: Mathematics #Functional Equations Stability Results #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2112.05955
openalex publication_date 2021/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The goal of this paper is to present a certain generalization of the classical Kontinuitätssatz of Behnke for holomorphic/meromorphic functions in terms of the lift to the envelope of holomorphy. We consider two non-equivalent formulations: "discrete" and "continuous" ones. Giving a proof of the "discrete" version we, somehow unexpectedly, construct a counterexample to the "continuous" one when convergence/continuity of analytic sets is considered in Hausdorff topology or, even in the stronger topology of currents. But we prove the "continuous" version of the Kontinuitätssatz if continuity is understood with respect to the Gromov topology. Our formulations seem to be not yet existing in the literature. A number of relevant examples and open questions is given as well.