Worn stones with flat sides all time regularity of the interface

We study the all time regularity of the free-boundary problem associated to the deformation of a compact weakly convex surface in ³, with a flat side, by its Gaussian Curvature. We show that under certain necessary regularity and non-degeneracy initial conditions the interface separating the flat from the strictly convex side, remains smooth on 0<t<T_c, up to the vanishing time T_c of the flat side.