Curvature and hyperbolicity of surfaces

Byung-Geun Oh, Purdue University

Abstract

An Aleksandrov surface is a generalization of two-dimensional Riemannian manifolds, and it is known by a theorem of A. Huber (1960) that every open simply connected Aleksandrov surface is conformally equivalent either to the unit disc (hyperbolic case) or to the plane (parabolic case). Thus one can study complex analysis on Aleksandrov surfaces, and in the first part of this thesis we prove a criterion for hyperbolicity of an Aleksandrov surface which has a nice tiling (or triangulation) and for which the negative curvature dominates. We apply this result to generalize a theorem of R. Nevanlinna and prove that a Riemann surface of class S is hyperbolic if negative excesses are spread uniformly over the corresponding Speiser graph. A partial answer for R. Nevanlinna's conjecture about Speiser graphs follows. In the second part of this thesis, we study the relations between linear isoperimetric inequalities and Gromov hyperbolicity on Speiser graphs, their duals, and the corresponding Riemann surfaces of class S, and show the following results: if a linear isoperimetric inequality holds for one of these three spaces, so does for the others; Gromov hyperbolicity of a Riemann surface of class S is equivalent to that of the corresponding dual Speiser graph; a linear isoperimetric inequality on a dual Speiser graph or the corresponding Riemann surface of class S implies Gromov hyperbolicity of them. We also construct some counterexamples so as to disprove the other implications.

Degree

Ph.D.

Advisors

Eremenko, Purdue University.

Subject Area

Mathematics

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