Katrin FAESSLER
Katrin FAESSLER
Katrin FAESSLER
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Extremality of the stretch II (maximal distortion)<br />
M4(Γ) = π 2 ln <br />
b −3<br />
a<br />
(cf. Korányi, Reimann) and thus<br />
and M4(fk(Γ)) = k −3 π 2 ln <br />
b −3<br />
a ,<br />
M4(fk(Γ)) K 2<br />
fk M4(Γ).<br />
Question<br />
Is the stretch map fk extremal for the maximal distortion within F?<br />
Proposition<br />
The stretch map fk is extremal for the maximal distortion in the class F0<br />
of C 1 orientation- and sphere-preserving maps g ∈ F which map the t-axis<br />
to the t-axis, i.e., Kfk ≤ Kg for all g ∈ F0.<br />
Proof idea: write contact transformation g ∈ F0 in logarithmic<br />
coordinates. Sphere-preserving: Ξ = Ξ(ξ), apply MVT: ∃ξ0 with<br />
Ξξ(ξ0) = k. Formula for distortion. Find p0 with K(p0, g) ≥ Kfk .<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 20 / 23
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