  For closed hyperbolic $3$-manifolds $M$ with volume less than a constant $V$,
we prove an inequality regarding the geometric $L^2$-norm and the topological
Thurston norm, which is qualitatively sharp and verifies a conjecture of Brock
and Dunfield in this case. Generically, we show that the $L^2$-norm is less
than a constant $c(V)$ times the Thurston norm by showing that any least area
closed surface is disjoint from the thin part.
  We then study the connection between the Thurston norm, best Lipschitz
circle-valued maps, and maximal stretch laminations, building on the recent
work of Daskalopoulos and Uhlenbeck, and Farre, Landesberg and Minsky. We show
that the distance between a level set and its translation is the reciprocal of
the Lipschitz constant, bounded by the topological entropy of the pseudo-Anosov
monodromy if $M$ fibers. For infinitely many examples constructed by Rudd, we
show the entropy is bounded from below by one-third the length of the
circumference.
