  We study, by Monte Carlo simulations, a coarse-grained model of a water
monolayer between hydrophobic walls at partial hydration, with a wall-to-wall
distance of about 0.5 nm. We analyze how the diffusion constant parallel to the
walls, $D_{\parallel}$, changes and correlates to the phase diagram of the
system. We find a locus of $D_{\parallel}$ maxima and a locus of
$D_{\parallel}$ minima along isotherms, with lines of constant $D_{\parallel}$
resembling the melting line of bulk water. The two loci of $D_{\parallel}$
extrema envelope the line of temperatures of density maxima at constant $P$. We
show how these loci are related to the anomalous volume behavior due to the
hydrogen bonds. At much lower $T$, confined water becomes subdiffusive, and we
discuss how this behavior is a consequence of the increased correlations among
water molecules when the hydrogen bond network develops. Within the
subdiffusive region, although translations are largely hampered, we observe
that the hydrogen bond network can equilibrate and its rearrangement is
responsible for the appearance of density minima along isobars. We clarify that
the minima are not necessarily related to the saturation of the hydrogen bond
network.
