Katrin FAESSLER

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“Logarithmic coordinates” (z, t) = where (ξ, ψ, η) ∈ R × (− π π 2 , 2 ) × R. i cos ψe ξ+i(ψ−3η) 2 , − sin ψe ξ , Simple description of Korányi spheres ξ = 2 ln ((z, t)H). Derived from coordinates on H 2 C which extend to ∂H2 C (Platis) ◮ η = 0: boundary of “standard flat pack” ◮ ψ = 0: plane C × {0} ◮ ξ = 0: Korányi unit sphere Related to the spherical coordinates (z, t) = (r cos1/2 θeiφ , r 2 sin θ) (Korányi, Reimann) via ξ = 2 ln r, ψ = −θ, η = 1 3 (π − θ − 2φ). Particular horizontal curves: ◮ s ↦→ (ξ(s), ψ(s), η(s)) = (ξ, s, η); ◮ s ↦→ (ξ(s), ψ(s), η(s)) = (s, ψ, η − <strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1 Picture tan ψ 3 s) (“radials”). Picture HCAA 2012 9 / 23
Figure: “Radials” connecting two Korányi spheres, each curve belongs to the Legendrian foliation of a Heisenberg cone associated to the parameter ψ.
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 27: “Logarithmic coordinates” (z, t
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 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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