... graph. (The existence of the curves σ1, γ1, γ2will be establishedin the next section.) First we need two lemmas. The first of these is the following:Lemma II.1.38. Given C1,ε0> 0, there ... σ2. The next lemma bounds the area of a minimal surface with two sides andan interior boundary in a small ball in terms of the length of the sides, providedthat the surface “initially leaves” the ... IIThen there exists a simple noncontractible curve ˇσ ⊂T1/32,1/4(σ) withLength(ˇσ)+ˇσ|kg|≤C1(1 + Area (T1/4(σ))) .(II.1.14)Proof. Following the first two steps of the proof of Lemma...