... thecurves γ∂jneeded for Corollary II.1.23.Corollary II.3.1. Given ω, m > 1, 1/4 ≥ ε>0, there exist Ω1,m0,δ sothat for r0,r2,R2,R0with 4Ω1r0≤ 4Ω1r2<R2<R0/(4 Ω1ω) ... subsectors:Corollary II.1.23. Given ω>8, 1 >ε>0, there exist m1, Ω1so thatthe following holds:Suppose γ ⊂ B2 r1∩ Σ is a curve with 1/(2 r1) <kg< 2/r1, Length(γ)=32 πm1r1, ... 0 <kg< 2/r1, anddistΣ(SR1(γ),∂Σ) ≥ r1/2 ,where R1> 2 r1. If there is a simple curve γ∂⊂ Σ with γ ⊂ γ∂, ∂γ∂⊂ ∂Σ,and soγx∩ γ∂= {x} for each x ∈ γ,then for...