Convex solutions to the power-of-mean curvature flow

TL;DR

This paper studies convex ancient solutions to the power-of-mean curvature flow for powers greater than 1/2, establishing estimates, asymptotic behaviors, and classification results for solutions in two dimensions.

Contribution

It provides new estimates for convex ancient solutions, describes their blow-down limits, and classifies convex ancient solutions in 2D for generalized curve shortening flow.

Findings

Blow-down of convex translating solutions converges to a specific power function.

Convex ancient solutions sweeping the plane must be shrinking circles.

Solutions not sweeping the plane are confined to a strip region.

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.