Convex solutions to the power-of-mean curvature flow

We prove some estimates for convex ancient solutions (the existence time for the solution starts at -∞) to the power-of-mean curvature flow, when the power is strictly greater than 1/2. As an application, we prove that in dimension two, the blow-down of an entire convex translating solution, namely u_h = 1/h u(h ^{1/1+α x}), locally uniformly converges to 1/1+α |x|^1+α as h → ∞. Another application is that for the generalized curve shortening flow (convex curve evolving in its normal direction with speed equal to a power of its curvature), if the convex compact ancient solution sweeps the whole space R², it must be a shrinking circle. Otherwise the solution must be defined in a strip region.