Katrin FAESSLER

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Choice of the curve family Γ0 Motivation: Curves in Γ0 play the role of radial segments (Korányi, Reimann), Γ0 and fk are related in the following way (these are general conditions which guarantee “=” in the modulus-distortion estimate.): ◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0: |((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)| For arbitrary f QC (diffeo) and γ horiz: |(fI ◦ γ) · | ≥ ||ZfI (γ)| − |ZfI (γ)|| · |˙z| and thus f ◦γ ρ′ dℓ ≥ γ ρ dℓ with ρ′ ρ = ||ZfI |−|¯ZfI || ◦ f −1 . With MSP: M4(f (Γ)) = infρ∈adm(Γ) Ω ′ ρ ′4 dµ = infρ∈adm(Γ) Ω ρ4K(·, f ) 2 dµ ◮ K(·, fk) is constant along each γψ,η ∈ Γ0; ◮ integration formula: 2 ln b h dµ = h(γψ,η(s))|˙zψ,η(s)| 4 ds 12 cos 2 (ψ)dψdη. A(a,b) Q 2 ln a
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)). <strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1 Explanation tan ψ 3 s), s ∈ [2 ln a, 2 ln b], HCAA 2012 19 / 23
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A stretch map on the Heisenberg gro
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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 51 and 52: Choice of the curve family Γ0 Moti
Page 53 and 54: Choice of the curve family Γ0 Moti
Page 55: 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
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Katrin FAESSLER

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