By assumption our points $u^i(z_i)$ don't run away along the semi-infinite ends attached to $\Sigma'$. The monotonicity lemma should now prevent the sequence of renormalized holomorphic discs from "thinning" out and having noncompact image. That is, our maps must use up at least a certain amount of energy for every ball whose center it passes through.
Chris Gerig
- 17.5k
- 2
- 71
- 116