β ζ Γ α (ζ) 2sec 1 α Figure I.2 The cone Γ α (ζ). P z (θ)

Transcription

1 z θ a igure I.1 The harmonic function θ(z). b z β ζ 2sec 1 α 0 Γ α (ζ) igure I.2 The cone Γ α (ζ). P z (θ) P z (θ) π θ 0 θ 0 π igure I.3 The function P z.

2 P z (θ) π 0 θ 0 θ n π igure I.4 Approximating P z by a step function. ζ α n σ n U n γ δn ϕ(γ δn ) β n igure I.5 The crosscuts γ δn and ϕ(γ δn ). z 1 z 2 0 x 1 a 1 a 2 igure I.6 Hyperbolic balls and geodesics. {T k } {S j } igure I.7 Whitney squares in D and Ω.

3 Ω k Ω j igure I.8 The domains Ω k are shaded. ϕ ψ k β x z ζ Ω k α ψ k (α) igure I.9 The map z = Φ(ζ) e iϕ 1 2πω 0 e iθ 2 ϕ z e iθ 1 e iϕ 2 igure I.10 Harmonic measure of an arc. l(q) Q e iθ 0 a Q l(q) igure I.11 A Carleson box.

4 I j T j I k T k igure I.12 Tents. a σ b a σ b igure II.1 The proof of Theorem 1.1. Γ 2 Ω Γ 3 Γ 1 ψ 3 ψ 2 ψ 1 Ω ψ 1 ψ 3 ψ 1 (Γ 2 ) ψ 1 (Γ 3 ) ψ 2 igure II.2 The proof of Lemma 2.2. Ω u = 0 V ζ u > 0 γ ψ v > 0 D v = 0 igure II.3 Straightening an analytic arc in Ω.

5 w 1 = (1 δ)z 1 w 2 = (1 δ)z 2 z 1 δ z 2 e iθ 1 e iθ 2 igure II.4 e i(θ+t) e iθ re i(θ+t) re iθ e i(θ t) re i(θ t) igure II.5 0 R igure III.1 Circular projection. K 0 K 1 K 2 K 3 K 4 K 5 igure III.2 The Cantor set.

6 J n δ ε 1 M M Ω n Ω n igure III.3 Proof of Proposition 5.2. A j+1 A j A j 1 J α I 2 j 2 (j 1) 2 (j 2) igure III.4 Annuli and the Cantor set. z z 0 D igure III.5 Circular Projection in D.

7 B(0, r) Ω 1 igure III.6 The proof of Corollary 9.3. U 1 2 igure III.7 Lower bound via reflection. z z igure III.8 Hall s lemma. γ Γ γ Γ γ Γ igure IV.1 Connecting and separating curves.

8 ζ 1 ζ 4 ϕ ϕ() ϕ() ζ 2 ζ 3 igure IV.2 xtremal distance in a quadrilateral. Ω = Ω Ω Ω Ω igure IV.3 xtension rule. γ Ω 1 Ω 2 G igure IV.4 Serial rule. 1 2 Ω 1 Ω 2 G 1 G 2 igure IV.5 Parallel rule.

9 γ Ω γ R T() T(Ω) γ T() igure IV.6 Symmetry rule. L 1 L 2 L 3 igure IV.7 Slit rectangle. ϕ ϕ() igure IV.8 xtremal distance and slit rectangles. ϕ() ψ ψ() ψ() igure IV.9 xtremal distance and slit annuli.

10 Ω z 0 σ igure IV.10 Distance from a point to an arc. σ 0 z f σ Ẽ 1 0 Ẽ 2 igure IV.11 Proof of Theorem 5.2. θ(x) Ω y = m(x) a x igure IV.12 A strip domain. b I x3 z 0 I x2 I x3 I x1 x 0 x 1 x 2 x 3 x 4 b R igure IV.13 Crosscut estimate of harmonic measure.

11 ζ I σ z I igure IV.14 J R 1 R 2 Ω 1 Ω 2 igure V.1 Teichmüller s Modulsatz. Γ 2 Γ 1 1 R R 2 σ 2ε Ω igure V.2 Ω, σ and their reflection about z = R.

12 Γ ε β (ζ) ζ β ε igure V.3 Truncated cone at ζ. Ω Γ α (0) i π 2 log(w) R Ω x δ x δ + S δ i π 2 0 igure V.4 Transforming the half-plane version to the strip version. i π 2 0 x n igure V.5 A comb region. x n i S ζ S U δ ψ 1 (U δ ) Reζ ψ igure V.6 Distortion near the ends of vertical crosscuts.

13 r s 0 igure V.7 Positive angular derivative at a tine. I s I r U 1 U 2 I t ε igure V.8 Image of a vertical crosscut. Ω Ω \ Ω M S Ω M Ω S igure V.9 M-Lipschitz subregion. Ω S U 1 U 1 s U U 4 U 4 U 2 t S U 3 Ω igure V.10 Proof of Sastry s lemma.

14 σ j v j z r j Ω S Ω z l j B j igure V.11 Proof of Lemma 6.3. ΩM Ω s γ 2 t γ 1 γ 3 igure V.12 Proof of Lemma 6.4. ζ 2sec 1 (α) 0 Γ h α h igure VI.1 The truncated cone Γ h α. K ϕ i U j U i K D K h igure VI.2 Cone domains U i.

15 w U T n (w) L n igure VI.3 n-lipschitz subregion. Ω γ k τ k V \ C \ U k σ k W \ L n L nl n igure VI.4 Nested regions in McMillan s theorem. Ω 0 Ω 1 Ω 2 Ω 5 igure VI.5 The von Koch snowflake. Ω W k wk Bk igure VI.6 Proof of Lemma 5.3.

16 z I(z) igure VI.7 I(z) consists of the base points of all cones Γ 2 containing z. α α z 0 σ igure VI.8 xtremal distance estimate of harmonic measure. α α T 3 j Jj 3 Ij 3 Jj 3 igure VI.9 irst generation towers. igure VI.10 Second generation towers.

17 (1 εt)e i(θ+t) e i(θ+t) e iθ Γ α (ζ) igure VII.1 T I 0 J z I igure VII.2 w 1 w w 2 w 1 w 2 w Quasicircle igure VII.3 Not Quasicircle Ω Ω δ w w 0 D δ (w) D δ (w) w w w 0 w w 1 0 L δ (w) L δ (w) δ w w 0 igure VII.4 The two possibilities in Condition (c).

18 7π 6 igure VII.5 xterior angle at a vertex. f z r f(z) m M igure VII.6 The quasiconformal image of a small circle. igure VII.7 Chord-arc curve. z 0 U Ω igure VII.8 Chord-arc subdomain.

19 z 0 John igure VII.9 Not John igure VII.10 Both sides John. D ν D ν ζ α 0 1,2 U 1 α 1,1 γ 2 β 1 w 0 σ γ 1 igure VIII.1 Proof of Lemma 2.5. A j (ζ ) θ Ω ζ ζ A 1 (ζ) igure VIII.2 Proof Lemma 3.3.

20 3t 1 t 1 t γ 3t K igure VIII.3 Known bounds for B(t). ϕ(ζ j ) ε δ ϕ(rζ j ) Ω igure VIII.4 a 2 a 3 Ω a 1 a 5 a 4 a 7 a 6 igure VIII.5 Polygonal tree. ib t ib igure VIII.6

21 ϕ R 1 Ω ϕ n ϕ 0 (H) igure VIII.7 B(a 1,2,δε 0 ) a 1 a 1 T a 0 a 2,2,2 a 2,1 a 0 a 2 a 2 B(a 2,ε 0 ) igure VIII.8 Dandelion construction. T n δ n a J U J f J f J (A J ) ζ J D \ V J A J igure VIII.9 The map f J near a J.

22 a 0 R igure VIII.10 L V n (J,3) W n (J,3) Q n (J,1) Q n 1 J Q n+1 (J,2,3) Q n (J,4) igure IX.1 Q n (J,2) L H K n (1,...,1) K n (1,...,4) W n (1,...,4) W igure IX.2 The proof of Lemma 1.3.

23 Ω 3 Ω 2 Ω 1 A 3 B 3 igure IX.3 A r r/α r igure IX.4 Qn+1,k Q n,j igure IX.5

24 A A Q Q Q Q B Ω Ω igure IX.6 Γ 1 Γ 2 K Γ 3 igure IX.7 h c U igure X.1 The special cone domain U = Γh β (ζ).

25 ζ ζ Γ α (ζ,u) U igure X.2 Γ α (ζ,u) and Γα(ζ,U). ζ U ζ z αd(z) Γ igure X.3 Γ = Γ α (ζ,u) is shaded. τ ζ c U igure X.4 The angle τ in (1.11).

26 U w z w z 0 c U igure X.5 U ζ Γ γ (ζ,u) Γ α (ζ) c U = 0 igure X.6 Cones satisfying (1.13). z t 2tβ (z,t) igure X.7

27 A 0 S 0 M(S) p A 1 S 1 S βl L igure X.8 Case 1. A 0 M(S) T A 1 S 1 L 1 βl S 0 L igure X.9 Case 2. Γ J n+1 2j J n+1 2j+1 J n j I n j igure X.10 ir/2 y = εx Γ r r igure X.11

28 z Q Q igure X.12 Q Q D(Q) Q igure X.13 Q J k Q k I igure X.14

29 L Γ 1 Γ 1 Γ 1 L z 1 B z 0 d 1 z 3 Γ 2 Γ j D j Case 1 Case 2 Case 3 igure X.15 ϕ(re iθ ) Ω ϕ(re iθ ) Ω W t (θ) igure X.16 On the left η β; on the right η = 1 while β is small. H 2 \ B 2 L 2 Ω H 1 A 1 z0 L 1 z0 Ω igure X.17 The region V 2.

30 D B 2 Γ Ω L 2 B 1 z 0 z 0 L 1 Γ Ω igure X.18 ϕ 1 (D) U J igure X.19 Q 1 Q 2 D(Q 2 ) Q 3 igure X.20 Q 1 L; Q 2,Q 3, M.

31 D(Q) Q z Q σ igure X.21 Γ ϕ 1 (Γ) Ω ϕ Γ Ω Ω igure X.22 j D igure X.23 The proof of (11.6).

32 Γ Ω j zj γ k Ω Γ Γ Ω Γ igure X.24 The proof of Theorem J J Ω J J J R τ(j) τ(ω) τ(j) τ(j) Ω d igure B.1. The doubled Riemann surface Ω d. W ε igure C.1

33 γ Ω p W igure C.2 K 1 K 2 K 3 igure D.1 z 0 z 1z2 z n igure.1 R 1 Ω 1 Ω 2 Ω 3 Ω 4 ϕ ϕ() R 2 R 3 ϕ() Ω 5 R 4 R 5 igure H.1 ϕ(ω j ) = R j. 0 1

34 z c ϕ = ω + i ω ϕ() ϕ() igure H.2 0 Reϕ(z c ) 1 ϕ ϕ() ϕ(z c ) ϕ() z c ϕ() ϕ() igure H.3 ϕ(z c ) 0 1 I x2 I x2 z 0 I x1 I x1 x 0 x 1 x 2 b R igure H.4 J J n Tn z n 1 igure I.1

35 C 0 w β ϕ(z 0 ) ϕ(σ) w w J 0 C n igure I.2 T n,j igure J.1 I k I j igure M.1 U 0 W 0 igure M.2