Highly Degenerate Harmonic Mean Curvature Flow

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.