Katrin FAESSLER

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Heisenberg stretch map Definition Let 0 < k < 1. The Heisenberg stretch map is given by Properties: fk(ξ, ψ, η) = (kξ, tan −1 tan ψ ( k ), η). fk is quasiconformal with fk(p)H = p k H , fk preserves C × {0}, moreover fk(z, 0) = (z|z| k−1 , 0), <strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1 HCAA 2012 12 / 23
Heisenberg stretch map Definition Let 0 < k < 1. The Heisenberg stretch map is given by Properties: fk(ξ, ψ, η) = (kξ, tan −1 tan ψ ( k ), η). fk is quasiconformal with fk(p)H = p k H , fk preserves C × {0}, moreover fk(z, 0) = (z|z| k−1 , 0), z f−1(z, t) = |z| 2 −t , −it |z| 4 +t2 is 1-QC inversion in the Korányi unit sphere with inversion relations dH(f−1(p), f−1(q)) = dH(p, q) pHqH <strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1 and f−1(p)H = 1 . pH HCAA 2012 12 / 23
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: 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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