Determining biholomorphic type of a manifold using combinatorial and algebraic structures

Sergiy A Merenkov, Purdue University

Abstract

We settle two problems of reconstructing a biholomorphic type of a manifold. In the first problem we use graphs associated to Riemann surfaces of a particular class. In the second one we use the semigroup structure of analytic endomorphisms of domains in [special characters omitted]. 1. We give a new proof of a theorem due to P. Doyle. The problem is to determine a conformal type of a Riemann surface of class Fq, using properties of the associated Speiser graph. Sufficient criteria of type have been given since 1930's when the class Fq was introduced. Also there were necassary and sufficient results which have theoretical value, but which are hard to apply. P. Doyle's theorem states that a non-compact Riemann surface of class Fq has a hyperbolic (parabolic) type, if and only if its extended Speiser graph is hyperbolic (parabolic). By a hyperbolic graph we mean a locally-finite infinite connected graph, which admits a non-constant positive superharmonic function with respect to the discrete Laplace operator. Otherwise a graph is parabolic. The usefulness of this criterion stems from the possibility of applying Rayleigh's short-cut method for graphs. We apply Doyle's theorem to give a counterexample to a conjecture of R. Nevanlinna that relates the type to an excess of a Speiser graph. More explicitely, the conjecture was that if the (upper) mean excess of a surface of class Fq is negative, then the surface is hyperbolic. We provide an example of a parabolic surface of class Fq with negative mean excess. 2. If there is a biholomorphic or antibiholomorphic map between two domains in [special characters omitted], then it gives rise to an isomorphism between the semigroups of analytic endomorphisms of these domains. Suppose, conversely, that we are given two domains in [special characters omitted] with isomorphic semigroups of analytic endomorphisms. Are they biholomorphically or antibiholomorphically equivalent? This question was raised by L. Rubel. Similar questions were studied in the setting of topological spaces. The case n = 1 was investigated by A. Eremenko, who showed that if we require that the domains are bounded, then the answer to the above question is positive. It was shown by A. Hinkkanen that the boundedness condition cannot be dropped. We prove that two bounded domains in [special characters omitted] with isomorphic semigroups of analytic endomorphisms are biholomorphically or antibiholomorphically equivalent. Moreover, we generalize this by requiring only the existence of an epimorphism between the semigroups.

Degree

Ph.D.

Advisors

Eremenko, Purdue University.

Subject Area

Mathematics

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