Katrin FAESSLER

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The setting: Heisenberg group Definition The Heisenberg group H 1 can be modeled on C × R such that the group law reads p ∗ p ′ = (z + z ′ , t + t ′ + 2 Im(zz ′ )), for p = (z, t), p ′ = (z ′ , t ′ ) ∈ H 1 with z, z ′ ∈ C, t, t ′ ∈ R. H 1 can be endowed with a left invariant metric where pH = (|z| 4 + t 2 ) 1 4 . For fixed s > 0, one has where δs(z, t) = (sz, s 2 t). dH(p, q) := q −1 ∗ pH, p, q ∈ H 1 , dH(δs(p), δs(q)) = s dH(p, q), p, q ∈ H 1 , <strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1 HCAA 2012 3 / 23
Objects of study: quasiconformal maps on (H 1 , dH) Definition (metric definition) A homeomorphism f : Ω → Ω ′ between domains in H 1 is called quasiconformal (QC) if there is 1 ≤ H < ∞ such that H(p, f ) := lim sup r→0+ <strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1 supdH(p,q)=r dH(f (p), f (q)) < H, for all p ∈ Ω. infdH(p,q)=r dH(f (p), f (q)) HCAA 2012 4 / 23
Page 1 and 2: A stretch map on the Heisenberg gro
Page 3 and 4: Radial stretch maps On the complex
Page 5: The setting: Heisenberg group Defin
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
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Extremality of the stretch I (mean
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Page 63 and 64:
Extremality of the stretch II (maxi
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Extremality of the stretch II (maxi
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References A. Korányi and H. M. Re
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Katrin FAESSLER

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