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

  1. K\ge 0 on M^+ and K\le 0 on M^-
  2. 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
  3. for each i, f^{-1}(\partial D_i) is a non-trivial 1-cycle in M
    (these are called top cycles)