Critical Points, the Gauss Curvature Equation and Blaschke Products

  • Daniela Kraus
  • Oliver RothEmail author
Part of the Fields Institute Communications book series (FIC, volume 65)


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 similar to those of Bergman space inner functions and they provide a natural generalization of the class of finite Blaschke products.


Blaschke products Elliptic PDEs Bergman spaces 

Mathematics Subject Classification

30H05 30J10 35J60 30H20 30F45 53A30 



The authors received support from the Deutsche Forschungsgemeinschaft (Grants: Ro 3462/3–1 and Ro 3462/3–2).

This paper is based on lectures given at the workshop on Blaschke products and their Applications (Fields Institute, Toronto, July 25–29, 2011). The authors would like to thank the organizers of this workshop, Javad Mashreghi and Emmanuel Fricain, as well as the Fields Institute and its staff, for their generous support and hospitality.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WürzburgWürzburgGermany

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