Discrete and Continuous Versions of the Continuity Principle

TL;DR

This paper generalizes the classical Kontinuit"atssatz for holomorphic functions, exploring discrete and continuous formulations, providing proofs, counterexamples, and establishing the version valid under Gromov topology.

Contribution

It introduces novel formulations of the Kontinuit"atssatz in terms of envelope of holomorphy and compares their validity across different topologies.

Findings

Counterexample in Hausdorff topology for continuous version

Proof of discrete version of the principle

Validation of continuous version under Gromov topology

Abstract

The goal of this paper is to present a certain generalization of the classical Kontinuit\"atssatz 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 Kontinuit\"atssatz 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.