Katrin FAESSLER

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Extremality of the stretch I (mean distortion) Proof (1): The family of “radial curves” Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η − satisfies K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)). Explanation tan ψ 3 s), s ∈ [2 ln a, 2 ln b], Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F? For Γ = {horiz. curves in A(a, b) connecting the two bdry components}: ρ0 ∈ adm(Γ) M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F. Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2ρ4 0 dµ. Thus, K(·, fk) 2 ρ 4 0 dµ ≤ K(·, g) 2 ρ 4 0 dµ for all g ∈ F. <strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1 HCAA 2012 17 / 23
Choice of the curve family Γ0 Motivation: Curves in Γ0 play the role of radial segments (Korányi, Reimann),
Page 1 and 2: A stretch map on the Heisenberg gro
Page 3 and 4: Radial stretch maps On the complex
Page 5 and 6: The setting: Heisenberg group Defin
Page 7 and 8: Objects of study: quasiconformal ma
Page 9 and 10: Horizontal structure on the Heisenb
Page 11 and 12: Contact property of QC maps Contact
Page 13 and 14: Definition of a Heisenberg stretch
Page 19 and 20: “Logarithmic coordinates” (z, t
Page 29 and 30: Figure: “Radials” connecting tw
Page 31 and 32: Contact transformations in logarith
Page 33 and 34: Contact transformations in logarith
Page 35 and 36: Heisenberg stretch map Definition L
Page 37 and 38: Heisenberg stretch map Definition L
Page 39 and 40: Analytic characterization of QC map
Page 41 and 42: Proposition The stretch map fk, 0 <
Page 43 and 44: Extremality of the stretch I (mean
Page 45 and 46: Modulus of curve families - a ‘qu
Page 49: Extremality of the stretch I (mean
Page 53 and 54: Choice of the curve family Γ0 Moti
Page 55 and 56: Choice of the curve family Γ0 Moti
Page 63 and 64: Extremality of the stretch II (maxi
Page 65 and 66: Extremality of the stretch II (maxi
Page 67 and 68: References A. Korányi and H. M. Re

Katrin FAESSLER

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