Critical sets of bounded analytic functions, zero sets of Bergman spaces and nonpositive curvature

TL;DR

This paper extends classical results about zero sets of bounded analytic functions to critical sets, linking them to weighted Bergman space zero sets and solving a geometric problem to characterize these critical sets.

Contribution

It establishes an analogue of Blaschke's theorem for critical sets, connecting complex analysis with differential geometry and weighted Bergman spaces.

Findings

Existence of indestructible Blaschke products with given critical sets

Critical sets of bounded analytic functions characterized as zero sets of weighted Bergman space ${\

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Abstract

A classical result due to Blaschke states that for every analytic self-map $f$ of the open unit disk of the complex plane there exists a Blaschke product $B$ such that the zero sets of $f$ and $B$ agree. In this paper we show that there is an analogue statement for critical sets, i.e. for every analytic self-map $f$ of the open unit disk there is even an indestructible Blaschke product $B$ such that the critical sets of $f$ and $B$ coincide. We further relate the problem of describing the critical sets of bounded analytic functions to the problem of characterizing the zero sets of some weighted Bergman space as well as to the Berger-Nirenberg problem from differential geometry. By solving the Berger-Nirenberg problem for a special case we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space ${\cal A}_1^2$.