Convex solutions to the power-of-mean curvature flow

Shibing Chen

Vol. 276 (2015), No. 1, 117–141

DOI: 10.2140/pjm.2015.276.117

Abstract

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

Keywords

mean curvature flow, convexity, translating solution, ancient solution

Mathematical Subject Classification 2010

Primary: 35J60

Milestones

Received: 19 February 2013

Revised: 19 October 2014

Accepted: 9 December 2014

Published: 1 July 2015