  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.
