Complex earthquakes are holomorphic.
Item
-
Title
-
Complex earthquakes are holomorphic.
-
Identifier
-
AAI3024833
-
identifier
-
3024833
-
Creator
-
Saric, Dragomir.
-
Contributor
-
Advisers: Linda Keen | Frederick P. Gardiner
-
Date
-
2001
-
Language
-
English
-
Publisher
-
City University of New York.
-
Subject
-
Mathematics
-
Abstract
-
Earthquakes on compact Riemann surfaces have been studied extensively. They are mappings of the Teichmuller space of a compact Riemann surface to itself. It is a result of Kerckhoff that an earthquake path is a real analytic path in the Teichmuller space of a compact surface. We give a generalization of Kerckhoff's result to the Teichmuller space of any Riemann surface, in fact, to the Universal Teichmuller Space.;We start from a bounded measure on the hyperbolic plane and the corresponding earthquake path parameterized by the positive real numbers. We extend the parameterization to a neighborhood of the real line in the complex plane. The extension is a holomorphic map in the parameter and, for a fixed parameter, it is a one to one map of the unit circle. Hence, the complex earthquake path, with the parameter in the given neighborhood of the real line, is a holomorphic motion of the unit circle. By Slodkowski's theorem, it is extendible to a holomorphic motion of the complex plane. Then, for a fixed positive parameter, the earthquake map is the restriction to the unit circle of a quasiconformal map of the complex plane preserving the unit disk. Thus an earthquake with a bounded measure is quasisymmetric. We also prove that a quasisymmetric earthquake has bounded measure.;The above results taken together show that for an earthquake the following are equivalent: (1) The measure of an earthquake is bounded, (2) An earthquake is quasisymmetric, (3) An earthquake path is a part of a holomorphic motion of the unit circle.;Moreover, an earthquake path with bounded measure is a real analytic path in the Universal Teichmuller Space.
-
Type
-
dissertation
-
Source
-
PQT Legacy CUNY.xlsx
-
degree
-
Ph.D.
