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

A classical result due to Blaschke states that for every analytic self-map f of the open unit disc of the complex plane there exists a Blaschke product B such that the zero sets of f and B agree. Indeed, a sequence is the zero set of an analytic self-map of the open unit disc if and only if it satisfies the simple geometric condition known as the Blaschke condition. In contrast, the critical sets of analytic self-maps of the open unit disc have not been completely described yet. In this paper, we show that for every analytic self-map f of the open unit disc 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 in a special case, we identify the critical sets of bounded analytic functions with the zero sets of the weighted Bergman space A(1)(2).