Katrin FAESSLER
Katrin FAESSLER
Katrin FAESSLER
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Extremality of the stretch I (mean distortion)<br />
Proof (1): The family of “radial curves”<br />
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −<br />
satisfies<br />
<br />
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).<br />
Explanation<br />
tan ψ<br />
3 s), s ∈ [2 ln a, 2 ln b],<br />
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?<br />
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:<br />
ρ0 ∈ adm(Γ)<br />
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.<br />
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2 ρ 4 0 dµ.<br />
<strong>Katrin</strong> Fässler (University of Helsinki) A stretch map on H 1<br />
HCAA 2012 17 / 23