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Journal of the London Mathematical Society Advance Access originally published online on December 13, 2007
Journal of the London Mathematical Society 2008 77(1):183-202; doi:10.1112/jlms/jdm095
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© 2007 London Mathematical Society

Critical points of inner functions, nonlinear partial differential equations, and an extension of Liouville's theorem

Daniela Kraus and Oliver Roth

Department of Mathematics
University of Würzburg
97074 Würzburg
Germany
roth{at}mathematik.uni-wuerzburg.de

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation (PDE) {Delta} u=4 e2u and combine this result with methods from nonlinear elliptic PDE to construct holomorphic maps with prescribed critical points and specified boundary behaviour. For instance, we show that for every Blaschke sequence {zj} in the unit disk there is always a Blaschke product with {zj} as its set of critical points. Our work is closely related to the Berger--Nirenberg problem in differential geometry.

2000 Mathematics Subject Classification 30D50, 35J65 (primary), 53A30, 30F45 (secondary).

The first author was supported by an HWP scholarship. The second author received partial support from the German--Israeli Foundation (grant G-809-234.6/2003).

Received November 7, 2006; revised June 22, 2007; published online December 13, 2007.


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