  We define the notion of fast finite-energy planes in the symplectization of a
closed 3-dimensional energy level $M$ of contact type. We use them to construct
special open book decompositions of $M$ when the contact structure is tight and
induced by a (non-degenerate) dynamically convex contact form. The obtained
open books have disk-like pages that are global surfaces of section for the
Hamiltonian dynamics. Let $S \subset \R^4$ be the boundary of a smooth,
strictly convex, non-degenerate and bounded domain. We show that a necessary
and sufficient condition for a closed Hamiltonian orbit $P\subset S$ to be the
boundary of a disk-like global surface of section for the Hamiltonian dynamics
is that $P$ is unknotted and has self-linking number -1.
