  We study the evolution of a weakly convex surface $\Sigma_0$ in $\R^3$ with
flat sides by the Harmonic Mean Curvature flow. We establish the short time
existence as well as the optimal regularity of the surface and we show that the
boundaries of the flat sides evolve by the curve shortening flow. It follows
from our results that a weakly convex surface with flat sides of class
$C^{k,\gamma}$, for some $k\in \mathbb{N}$ and $0 < \gamma \leq 1$, remains in
the same class under the flow. This distinguishes this flow from other,
previously studied, degenerate parabolic equations, including the porous medium
equation and the Gauss curvature flow with flat sides, where the regularity of
the solution for $t >0$ does not depend on the regularity of the initial data.
