Characterization of Quasiconformal Mappings and Extremal Length Decomposition Open Access
Zou, Wenfei (2014)
Published
Abstract
Quasiconformal mappings have abundant subtle analytic and geometric properties, which can be used widely in various contexts. The reason probably lies in that there exists several equivalent definitions for quasiconformal mappings. While conformal mappings preserve measures of angles, quasiconformal mappings are their natural generalizations. Geometrically, a quasiconformal mapping maps infinitesimal balls to infinitesimal ellipsoids with uniformly controlled eccentricity in space. This suggests that it is reasonable to use measures of angles to characterize quasiconformal mappings. In the first part of this dissertation, a measure of angle called topological angle is used to characterize quasiconformal mappings in higher dimensional Euclidean space, generalizing a similar result in the plane.
The second part of the dissertation deals with some important conformal invariants in the study of geometric function theory, such as quasiextremal distance (or QED) constant and extremal length. QED domains are a class of domains closely connected to quasiconformal mapping theory. The QED constant is a naturally defined conformal invariant on a domain whose values reflect the geometry of a domain. In this part, a sharp upper bound for the QED constant in terms of boundary dilatation is obtained for a finitely connected domain on the complex plane. Furthermore, the extremal length (or its reciprocal called modulus) of a curve family plays an essential role in studying quasiconformal mappings. In the second part of this dissertation, a decomposition result is established for the extremal length of a curve family in a finitely connected domain. This can be regarded as a natural generalization of subadditivity of extremal length. It is also a key ingredient in obtaining the sharp upper bound for the QED constant mentioned above.
Table of Contents
1 Introduction .......... 1
1.1 Quasiconformal mappings ........... 1
1.2 The modulus of a curve family............... 4
1.2.1 The definition of modulus ......... 4
1.2.2 Properties of modulus ............ 5
1.3 QED Domain and QED Constant............... 6
1.4 Outline and summary of results ............... 8
2 Characterization of quasiconformal mappings using topological angle ...9
2.1 Introduction ............................ 9
2.2 Definition of topological angle .................. 10
2.3 Topological angle under linear mappings ............ 10
2.4 Topological angle under differentiable homeomorphism ........ 15
2.5 Lower bound of topological angles under quasiconformal mappings......... 18
2.5.1 Quasisymmetric mappings .............. 18
2.5.2 Lower bound of angles under quasiconformal mappings .......... 18
2.6 Characterization of quasiconformal mappings by topological angles in R3 .......... 20
2.6.1 Preliminary ........................ 20
2.6.2 Characterization Theorem ................ 21
2.6.3 Derivative of measure................... 22
2.6.4 f is ACL on G ...................... 23
2.6.5 fz is ACL3 on G ..................... 28
2.6.6 Completion of proof of Characterization Theorem 2.9 ..... 29
2.7 Generalization to Rn+1 ...................... 30
3 QED reflection constant ...................34
3.1 Fundamental properties about the QED reflection constant ......... 35
3.1.1 Properties about the QED reflection constant .............. 35
3.2 The QED reflection constant for smooth domains ........... 37
3.2.1 Preliminary ....... 37
3.2.2 The QED reflection constant............. 39
4 Decomposition of extremal length on finitely connected domains ................44
4.1 Reduction ...................... 45
4.2 Preliminaries ....................... 45
4.3 Mixed DirichletNewmann problem ................... 46
4.4 Critical points.................... 47
4.5 Level curves and critical points ................... 48
4.6 No interior critical points ...................... 49
4.7 Domain Decomposition ...................... 51
4.8 Integration on critical level curves ............... 52
4.9 Completion of the Proof ....................... 54
5 The QED constant and the Boundary dilatation on multiplyconnected domains .........57
5.1 Boundary dilatation of multiplyconnected domains ..................... 58
5.2 Two Lemmas .................................. 60
5.3 Proof of Main Theorem ............................................... 65
Bibliography .........................................68
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