Harmonic Measure, Reduced Extremal Length and Quasicircles Open Access

Shi, Huiqiang (2016)

Permanent URL: https://etd.library.emory.edu/concern/etds/jm214p883?locale=en
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Abstract

It is well known that there is a close connection between the analytic behavior of the sewing homeomorphism induced by a Jordan curve and the geometry of this Jordan domain. For example, the sewing homeomorphism is quasisymmetric if and only if the Jordan domain is a quasidisk. This dissertation is devoted to the further study of this type of connection. Several equivalent conditions are established for sewing homeomorphism to be bi-Lipschitz or bi-Holder. In particular, we explore these conditions by using conformal invariants such as harmonic measure, extremal distance and reduced extremal distance. Furthermore, some parallel conditions for a quasicircle are obtained.

Table of Contents

1 Introduction ......................................................................................... 1

1.1 Extremal length................................................................................... 1

1.2 Modulus............................................................................................ 4

1.3 Extremal distance................................................................................ 5

1.4 Quasiconformal mapping...................................................................... 6

1.5 Quasicircle......................................................................................... 8

1.6 Riemann mapping theorem and Schwarz-Christoffel formula.....................8

1.7 Reduced extremal distance.................................................................... 9

2 Conformal invariants ............................................................................ 11

2.1 Extremal domains for modulus ........................................................... 11

2.1.1 Grotzsch extremal domain............................................................... 11

2.1.2 Teichmuller extremal domain .......................................................... 12

2.1.3 Mori extremal domain .................................................................... 12

2.2 Estimate of extremal distance in the unit disk........................................15

2.3 Comparision of extremal distance and reduced extremal distance ............18

3 Equivalent conditions for Bi-Lipschitz sewing homeomorphism .................. 24

3.1 Harmonic measure ............................................................................ 24

3.2 Equivalent conditions for hΩ to be bi-Lipschitz homeomorphism ..............25

3.3 Proof of theorem 3.2.1 ........................................................................ 27

4 Equivalent conditions for bi-Holder sewing homeomorphism ......................36

4.1 Equivalent conditions for hΩ to be bi-Holder homeomorphism ................ 36

4.2 Proof of theorem 4.1.2 ........................................................................ 38

5 Harmonic measure property and quasicircle ........................................... 47

5.1 Preliminary ..................................................................................... 47

5.2 Equivalent conditions for a quasicircle .................................................... 49

5.3 Proof of theorem 5.2.1 ........................................................................ 50

5.4 HMPandquasicircle ........................................................................... 52

6 Characterization of unit circle by using Robin Capacity ............................. 56

6.1 Robin Function and Robin Capacity ......................................................... 56

6.2 Characterization of unit circle ............................................................... 58

7 Future work ........................................................................................ 63

Bibliography .......................................................................................... 66

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