... Rk and Rk,j≥ Rk,j. Similarly, each rxcan be replaced by any rx≤ rx. This will beused freely in the proof of Theorem 0.1 below.Proof of Theorem 0.1. Let xk,gk,Rk,Rk,j and ... curvature and area of stable sectors. A stable sector in the sense of [CM3], [CM4] is a stable subset of a minimal surface given as half of a normal tubular neighborhood (in thesurface) of a strictly ... ≤ g<∞ and Si→∞.After going to a subsequence,Σj, and possibly replacing Sjby Rj and ΣjbyΣ0,j,Rjwhere R0≤ Rj≤ Sj and Rj→∞, thengen(Σj,0,R0) = gen(Σj) ≤ g and either...