Analytic iterations on Riemann surfaces

A complex analytic family of mappings P → M(α, P) from an abstract Riemann surface (analytic manifold) into itself is studied. The mapping M(α, P) is assumed to satisfy in local coordinates the autonomous differential equation = L(w), and the condition M(O, P) = P. Under certain assumptions of regularity of the reciprocal differential L in a domain D ⊂ S, we prove that for every fixed α, ∣a∣ < α, the mapping M(α, P) is conformal and one to one in D. Moreover, it is shown that the family of mappings M(α, P) satisfies the iteration equation M[a, M(b, P)] = M(a + b, P) and hence is an analytic group (analytic iteration).

Boundary Components of Riemann Surfaces

The boundary components of an abstract Riemann surface were defined by B. v. Kérékjértó [7] and utilized in the book [14] written by S. Stoïlow. It is the purpose of the present paper to investigate their images under conformal mapping and to solve the Dirichlet problem with boundary values distributed on them.

Note on the Harmonic Measure of the Accessible Boundary of a Covering Riemann Surface

The following relation was set up in [5] for an open covering Riemann surface ℜ with positive boundary over an abstract Riemann surface (1) when the universal covering surface of the projection is not of hyperbolic type when is of hyperbolic type this relation is reduced to(2)

On a Covering Surface over an Abstract Riemann Surface

1. Let be an abstract Riemann surface in the sense of Weyl-Radó, and an open covering surface over . If a curve C = {P(t);0≦t<1} on tends to the ideal boundary of but its projection terminates at an inner point of as t→1, we shall say that C determines an accessible boundary point (which will be abbreviated by A.B.P.) of relatively to . The set of all the A.B.P.S of relative to will be called accessible boundary (relative to ) and denoted by 3i() or by (, ). Throughout in this paper () will be supposed to be non-empty.

Dirichlet Problems on Riemann Surfaces and Conformal Mappings

The object of this paper is an investigation of existence problems and Dirichlet problems on an abstract Riemann surface in the sense of Weyl-Radó or on a covering surface over it, and of boundary correspondence in the conformal mapping of the surface.