Critical points, the Gauss curvature equation and Blaschke products

TL;DR

This survey explores the characterization of critical points of bounded analytic functions in the unit disk, linking complex analysis, differential geometry, and Bergman space zero sets, highlighting maximal Blaschke products.

Contribution

It introduces the concept of maximal Blaschke products associated with critical points, generalizing finite Blaschke products and connecting multiple mathematical areas.

Findings

Existence of a unique maximal Blaschke product for each set of critical points

Maximal Blaschke products share properties with Bergman space inner functions

These products generalize finite Blaschke products to broader classes

Abstract

In this survey paper, we discuss the problem of characterizing the critical sets of bounded analytic functions in the unit disk of the complex plane. This problem is closely related to the Berger-Nirenberg problem in differential geometry as well as to the problem of describing the zero sets of functions in Bergman spaces. It turns out that for any non-constant bounded analytic function in the unit disk there is always a (essentially) unique "maximal" Blaschke product with the same critical points. These maximal Blaschke products have remarkable properties simliar to those of Bergman space inner functions and they provide a natural generalization of the class of finite Blaschke products.