1549055259-Ubiquitous_Quasidisk__The__Gehring_

2.4. LINEAR LOCAL CONNECTIVITY 29 c v u FIGURE 2.4 By performing a final pair of similarity mappings we may assume that L' as we ...

30 2. GEOMETRIC PROPERTIES E' FIGURE 2.5 THEOREM 2.4.7 (Gehring [ 48 ], Walker [165]). The image of a c-lin early lo- cally conn ...

2.5. DECOMPOSITION D and rr--tr as j----too. Hence if z E Bo = B (z 0 , r), then for large j lz - Wjl::; lz - zol + lzo - Wjl &l ...

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CHAPTER 3 Conformal invariants We consider here eight conformally inva riant descriptions for a quasidisk D. Four of these compa ...

34 3. CONFORMAL INVARIANTS conformal type of I: is determined by N= 2m+n-3 real numbers. The three cases where N = 1, together w ...

3.3. BOUNDS FOR HYPERBOLIC DISTANCE 35 where the infimum is taken over all rectifiable curves (3 which join z 1 and z 2 in D. Ag ...

36 3. CONFORMAL INVARIANTS LEMMA 3.3.3. The Apollonian metric aD is a Mobius invariant pseudo-metric in D. It is a metric whenev ...

3.3. BOUNDS FOR HYPERBOLIC DISTANCE 37 Next let 'Y be the hyperbolic segment joining z 1 and z 2 in D. Then 1 1. ----,---------, ...

38 3. CONFORMAL INVARIANTS where w 1 = r e-i et./^2 and w 2 = r ei et./^2 , and with (3.2.2) we obtain 1 1 112 an(z ( a. -^2 e a ...

3.5. UNIFORM DOMAINS 39 3.4. Geometry of hyperbolic segments A second useful characterization for quasidisks is an analogue of t ...

40 3. CONFORMAL INVARIANTS DEFINITION 3.5.l. An arbitrary domain D C R^2 is uniform if there exists a constant c ~ 1 such that e ...

3.6. MIN-MAX PROPERTY OF HYPERBOLIC SEGMENTS 41 by (3.5.7) and we obtain min length(lj)::; a(c + 1) length ( a)::; b(c + 1) dist ...

42 3. CONFORMAL INVARIANTS Then with (3.6.4) and (3 .6.5) we obtain and (1 + lf(w)l^2 ) (1 + lf(oo)l2) ~ (rilf(oo)l^2 + r~)lf(w) ...

3.8. HARMONIC QUASISYMMETRY 43 For example, if D is the right half-plane and/ is the arc in fJD with endpoints i a, i b where 0 ...

44 3. CONFORMAL INVARIANTS is sense-preserving with ¢(1) = l. We show first that ¢(z) = z. For this let 11 and 12 be the upper a ...

3.8. HARMONIC QUASISYMMETRY 45 for j = 1, 2, 3 and we obtain 1 ¢ w(l,11;D) = -(arctana~ - arctanaf) = -, n n 1 ¢ w(l,12;D) = -(a ...

46 3. CONFORMAL INVARIANTS 3.9. Harmonic bending 2 Suppose that 'Y is a closed arc with endpoints z 1 and z 2. Then D = R \ 'Y ...

3.10. QUADRILATERALS 47 i a+i R f 0 a FIGURE 3.2 The modulus of a quadrilateral Q can also be given in terms of the modulus of t ...

48 3. CONFORMAL INVARIANTS a strictly decreasing homeomorphism of (0, 1) onto (0, oo). Furthermore, we have that 4 μ(r) '="log - ...