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
)
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