Construction Theorem

 If f\colon M\to\R^3 is a tight immersion, then M can be decomposed into two regions, M^+ and M^- where K\ge 0 on M^+ and K\le 0 on M^- f embeds M^+ onto the complement in \dHf of a finite number of disjoint planar convex closed disks D_1,\ldots,D_k in \dHf for each i, f^{-1}(\partial D_i) is a non-trivial 1-cycle in M (these are called top cycles)