Uniqueness of entire graphs evolving by mean curvature flow

In this paper we study the uniqueness of graphical mean curvature flow with locally Lipschitz initial data. We first prove that rotationally symmetric entire graphs are unique, without any further assumptions. Our methods also give an alternative simple proof of uniqueness in the one-dimensional case. In the general case, we establish the uniqueness of entire proper graphs that satisfy a uniform lower bound on the second fundamental form. The latter result extends to initial conditions that are proper graphs over subdomains of ℝ n {\mathbb{R}^{n}} . A consequence of our result is the uniqueness of convex entire graphs, which allow us to prove that Hamilton’s Harnack estimate holds for mean curvature flow solutions that are convex entire graphs.