  We consider the geometric evolution problem of entire graphs moving by
fractional mean curvature. For this, we study the associated nonlocal
quasilinear evolution equation satisfied by the family of graph functions. We
establish, using an analytic semigroup approach, short time existence,
uniqueness and optimal H\"older regularity in time and space of classical
solutions of the nonlocal equation, depending on the regularity of the initial
graph. The method also yields $C^\infty-$smoothness estimates of the evolving
graphs for positive times.
