Leo Sario

Parabolicity and the Riemann Theorem

Consider a 2-dimensional vector field α of class C 1 defined on the punctured closed unit disk \( \bar \Omega :0 < \left| z \right| \leq 1 \); here α may or may not have a singularity at z = 0. We view the unit circle |z| =1 as the relative boundary ∂Ω of the punctured unit disk Ω: 0 < |z| < 1 and the origin z = 0 as the ideal boundary of Ω. We are interested in the ‘behavior at the boundary’ z = 0 of ‘harmonic functions’ u on Ω defined by the elliptic equation L α u = Δu + α · ∇u = 0 on Ω. We say that α is parabolic if sup Ω u = max ∂Ω u for every bounded solution u of L α u = 0 on Ω continuous on \(\bar \Omega \). We also say that the Riemann theorem is valid for α if lim z→0 u (z) exists for every bounded solution u of L α u = 0 on Ω. The present study is motivated by the feeling that there must be a close connection between these two properties. However, we shall show that, contrary to this expectation, parabolicity has nothing to do with the Riemann theorem.

Capacity, Stability, and Extremal Length

We begin by considering a harmonic function whose boundary behavior is a combination of L0- and L1-type behavior. This leads to a generalization of harmonic measure and capacity. In §2 we relate this function to some extremal length problems. The remaining sections treat applications to conformal mapping and stability problems.

Introduction: What are Principal Functions?

The existence of certain harmonic functions with prescribed singularities and prescribed boundary behavior is central to a variety of areas of complex function theory. It provides the unifying factor for all the topics of this book. Such functions will be called principal functions. For further orientation let us consider how they are constructed and some of their uses.

Principal Functions on Harmonic Spaces

In this chapter we shall discuss the problem of finding on a given harmonic space a harmonic function which imitates the behavior of a given harmonic function on a neighborhood of the ideal boundary of the harmonic space.

Prerequisite Riemann Surface Theory

In this chapter we list the basic properties of Riemann surfaces which are needed in the remainder of the book. For the most part we have omitted detailed proofs here since the reader will often be able to supply his own or else refer to standard textbooks for them.

The Normal Operator Method

In this chapter the basic tools are created which will be used throughout the remainder of the book. The central topic is the Main Existence Theorem for principal functions, given in §1. The hypotheses of this theorem require the existence of normal operators. That such operators always exist is a nontrivial fact; its proof is given in §2 by constructing the operators L0 and L1 on an arbitrary Riemann surface. The method used there, which is typical of such problems, consists of constructing operators on compact bordered subregions and passing to a limit.

Principal Forms and Fields on Riemannian Spaces

Having discussed various aspects of the normal operator method on Riemann surfaces, and consequently on 2-dimensional Riemannian spaces, we now turn to their counterparts on higher dimensional Riemannian spaces. As far as functions, i.e. 0-forms, are concerned, it is plausible that all results thus far discussed, save those involving complex analyticity, can be reproduced for higher dimensions. That this is actually the case is shown in §1.

Integrated forms derived from nonintegrated forms of value distribution theorems under analytic and quasi-conformal mappings

We consider complex analytic or, more generally, quasi-conformal mappings into closed Riemann surfaces W 0 of open Riemann surfaces W p that carry capacity functions with compact level lines. We shall show that, for these mappings, the so-called integrated forms of value distribution theorems can be derived from the nonintegrated forms.

Mappings into Open Riemann Surfaces

Our considerations thus far have been limited to the case of closed range surfaces. We now drop this restriction and study given complex analytic mappings of arbitrary (open) Riemann surfaces into arbitrary Riemann surfaces.

Functions on Parabolic Riemann Surfaces

In I.10 to 12 we encountered examples of R p -surfaces, i.e., surfaces possessing capacity functions with compact level lines. We shall now draw such surfaces under a more systematic study and show that every parabolic surface is of type R p .