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

TL;DR

This paper extends Liouville's theorem for a specific PDE, enabling the construction of holomorphic maps with prescribed critical points and boundary conditions, linking nonlinear PDEs with complex analysis and differential geometry.

Contribution

It generalizes Liouville's theorem for a nonlinear PDE and introduces methods to construct holomorphic maps with designated critical points and boundary behavior.

Findings

Existence of Blaschke products with prescribed critical points.

Extension of Liouville's theorem to solutions of $ riangle u=4 e^{2u}$.

Connection between nonlinear PDEs and complex geometric problems.

Abstract

We establish an extension of Liouville's classical representation theorem for solutions of the partial differential equation $\Delta u=4 e^{2u}$ 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.