2007/08/29 by Daniela Kraus, Oliver Roth
Mathematics · #Analytic and geometric function theory #Blaschke product #Elliptic partial differential equation #Extension (predicate logic) #First-order partial differential equation #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory #Holomorphic function #Mathematical analysis #Mathematics #Nonlinear system #Partial derivative #Partial differential equation #Pure mathematics #Sequence (biology) #Unit disk #math.AP #math.CV #msc:30D50 #msc:30F45 #msc:35J65 #msc:53A30
paper · pdf · doi:10.1112/jlms/jdm095
published as J. London Math. Soc. 77 No. 1, 183-202, 2008 · 21 pages
arxiv created 2007/08/29 · openalex publication_date 2007/12/13 · arxiv updated 2014/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation (PDE) Δ 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.