  We consider the standard first passage percolation model on $\mathbb Z^ d$
with a distribution $G$ taking two values $0<a<b$. We study the maximal flow
through the cylinder $[0,n]^ {d-1}\times [0,hn]$ between its top and bottom as
well as its associated minimal surface(s). We prove that the variance of the
maximal flow is superconcentrated, i.e. in $O(\frac {n^{d-1}} {\log n})$, for
$h\geq h_0$ (for a large enough constant $h_0=h_0(a,b)$). Equivalently, we
obtain that the ground state energy of a disordered Ising ferromagnet in a
cylinder $[0,n]^ {d-1}\times [0,hn]$ is superconcentrated when opposite
boundary conditions are applied at the top and bottom faces and for a large
enough constant $h\geq h_0$ (which depends on the law of the coupling
constants). Our proof is inspired by the proof of Benjamini--Kalai--Schramm.
Yet, one major difficulty in this setting is to control the influence of the
edges since the averaging trick used in the proof of Benjamini--Kalai--Schramm
fails for surfaces. Of independent interest, we prove that minimal surfaces (in
the present discrete setting) cannot have long thin chimneys.
