  We consider stochastic spin-flip dynamics for: (i) monotone discrete surfaces
in Z^3 with planar boundary height and (ii) the one-dimensional discrete
Solid-on-Solid (SOS) model confined to a box. In both cases we show almost
optimal bounds O(L^2polylog(L)) for the mixing time of the chain, where L is
the natural size of the system. The dynamics at a macroscopic scale should be
described by a deterministic mean curvature motion such that each point of the
surface feels a drift which tends to minimize the local surface tension.
Inspired by this heuristics, our approach consists in bounding the dynamics
with an auxiliary one which, with very high probability, follows quite closely
the deterministic mean curvature evolution. Key technical ingredients are
monotonicity, coupling and an argument due to D.B. Wilson in the framework of
lozenge tiling Markov Chains. Our approach works equally well for both models
despite the fact that their equilibrium maximal height fluctuations occur on
very different scales (logarithmic for monotone surfaces and L^{1/2} for the
SOS model). Finally, combining techniques from kinetically constrained spin
systems together with the above mixing time result, we prove an almost
diffusive lower bound of order 1/L^2 up to logarithmic corrections for the
spectral gap of the SOS model with horizontal size L and unbounded heights.
