Katrin FAESSLER

Extremality of the stretch I (mean distortion)
Proof (1): The family of “radial curves”
Γ0 := {γψ,η}, where γψ,η(s) = (s, ψ, η −
satisfies
K(·, fk) 2ρ4 0 dµ = M4(fk(Γ0)).
Explanation
tan ψ
3 s), s ∈ [2 ln a, 2 ln b],
Proof (2): How are M4(fk(Γ0)), M4(g(Γ0)) related for arbitrary g ∈ F?
For Γ = {horiz. curves in A(a, b) connecting the two bdry components}:
ρ0 ∈ adm(Γ)
M4(fk(Γ0)) ≤ M4(g(Γ)) ∀g ∈ F.
Apply the modulus inequality M4(g(Γ)) ≤ K(·, g) 2ρ4 0 dµ.
Thus,
K(·, fk) 2 ρ 4
0 dµ ≤ K(·, g) 2 ρ 4 0 dµ for all g ∈ F.
Katrin Fässler (University of Helsinki) A stretch map on H 1
HCAA 2012 19 / 23