The structure of extremal domains associated with an analytic function

Abstract

In [11] and [12] we investigated two extremal problems that are connected with the analytic continuation of an analytic function f(z). In this paper we show that the complement K 0 of the uniquely existing maximal domain for single-valued analytic continuation of f(z) in has a special structure: :K 0 consists of the union of analytic arcs with a certain compact set E 0, where E 0 defined by the property that at every boundary point of E 0 the analytic continuation of f(z) has a singularity. The analytic arcs in K 0 ˜ E 0 are trajectories of a quadratic differential. If the function f(z) is algebraic, then the set E 0 is finite and the quadratic differential specially simpie.