Convex solutions to the power-of-mean curvature flow

Abstract

We prove some estimates for convex ancient solutions (the existence time for the solution starts from $-\infty$) to the power-of-mean curvature flow, when the power is strictly greater than 1/2. As an application, we prove that in two dimension, the blow-down of the entire convex translating solution, namely $u_{h}=\frac{1}{h}u(h^{\frac{1}{1+\alpha}}x),$ locally uniformly converges to $\frac{1}{1+\alpha}|x|^{1+\alpha}$ as $h\rightarrow\infty$. Another application is that for 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 $\textbf{R}^{2}$, it it has to be a shrinking circle. Otherwise the solution is defined in a strip region.