Katrin FAESSLER

Choice of the curve family Γ0
Motivation:
Curves in Γ0 play the role of radial segments (Korányi, Reimann),
Γ0 and fk are related in the following way (these are general conditions
which guarantee “=” in the modulus-distortion estimate.):
◮ “minimal stretching property” (MSP): For γψ,η = γ = (z, t) ∈ Γ0:
|((fk)I ◦ γ) · (s)| = ||Z(fk)I (γ(s))| − |Z(fk)I (γ(s))|| · |˙z(s)|
For arbitrary f QC (diffeo) and γ horiz: |(fI ◦ γ) · | ≥ ||ZfI (γ)| − |ZfI (γ)|| · |˙z| and
thus
f ◦γ ρ′ dℓ ≥
γ ρ dℓ with ρ′ ρ
=
||ZfI |−|¯ZfI || ◦ f −1 .
With MSP: M4(f (Γ)) = infρ∈adm(Γ) Ω ′ ρ ′4
dµ = infρ∈adm(Γ) Ω ρ4K(·, f ) 2 dµ
◮ K(·, fk) is constant along each γψ,η ∈ Γ0;