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 $h\to \infty$. 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 ${\text{R}}^2$, it it has to be a shrinking circle. Otherwise the solution is defined in a strip region.

Some details are compressed