  We prove that the $L_4$ norm of the vertical perimeter of any measurable
subset of the $3$-dimensional Heisenberg group $\mathbb{H}$ is at most a
universal constant multiple of the (Heisenberg) perimeter of the subset. We
show that this isoperimetric-type inequality is optimal in the sense that there
are sets for which it fails to hold with the $L_4$ norm replaced by the $L_q$
norm for any $q<4$. This is in contrast to the $5$-dimensional setting, where
the above result holds with the $L_4$ norm replaced by the $L_2$ norm.
  The proof of the aforementioned isoperimetric inequality introduces a new
structural methodology for understanding the geometry of surfaces in
$\mathbb{H}$. In previous work (2017) we showed how to obtain a hierarchical
decomposition of Ahlfors-regular surfaces into pieces that are approximately
intrinsic Lipschitz graphs. Here we prove that any such graph admits a foliated
corona decomposition, which is a family of nested partitions into pieces that
are close to ruled surfaces.
  Apart from the intrinsic geometric and analytic significance of these
results, which settle questions posed by Cheeger-Kleiner-Naor (2009) and
Lafforgue-Naor (2012), they have several noteworthy implications, including the
fact that the $L_1$ distortion of a word-ball of radius $n\ge 2$ in the
discrete $3$-dimensional Heisenberg group is bounded above and below by
universal constant multiples of $\sqrt[4]{\log n}$; this is in contrast to
higher dimensional Heisenberg groups, where our previous work showed that the
distortion of a word-ball of radius $n\ge 2$ is of order $\sqrt{\log n}$.
