Construction of open up mappings with rational functions and related questions

Using tools from algebraic geometry and the theory of Riemann surfaces, we establish the existence of special conformal mappings. Special emphasis is put on a constructive approach, and these mappings are rational functions with minimal degree. Three problems are discussed: the existence of a rational open up mapping, the critical value problem, and the critical point problem. We discuss the relations between the three problems, and recollect related questions which are scattered in the literature. Moreover, we investigate the properties of a given rational function as an open up mapping with the theory of quadratic differentials. This is joint work with Sergei Kalmykov (SJTU) and Olivier Sete (TU Berlin) and based on arxiv preprint https://arxiv.org/abs/1909.00264