The mean curvature flow of a surface given by a graph $X : B \subset \Bbb{R}^n \to [0,\infty)$ is given by $$ X_t (x,t) = H(x,t) \vec n(x,t) $$ where $H$ is the mean curvature and $\vec n$ is the normal vector. A result of Ecker and Huisken says that if the initial condition $X_0$ is Lipschitz then for small $t$ the surface determined by $X_t$ is smooth.

Another property stated often in texts speaking about mean curvature flows is the fact that if the initial surface $M_0$ is mean-convex (i.e. it has non-negative curvature everywhere) then $M_t$ is also mean-convex.

My question is:

Is there any condition that we can impose on an initial condition which is Lipschitz, but not $C^2$ (and therefore the curvature is not well defined) so that $M_t$ has positive curvature for $t>0$?


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