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

Abstract

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\Delta u=4e^{2u}$ 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 $\{z_j\}$ in the unit disk there is always a Blaschke product with $\{z_j\}$ as its set of critical points. Our work is closely related to the Berger-Nirenberg problem in differential geometry.

21 pages

Cited by in corpus (5)

• Maximal Blaschke Products
• Beurling's free boundary value problem in conformal geometry
• The Jacobian Conjecture for the space of all the inner functions
• Minimal degree rational open up mappings and related questions
• Finite Blaschke products with prescribed critical points, Stieltjes polynomials, and moment problems