Harmonic Measure, Reduced Extremal Length and Quasicircles Open Access

Shi, Huiqiang (2016)

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


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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