Convex Solutions to the Power-of-mean Curvature
speaker: Shibing Chen (University of Toronto)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 \(\frac{1}{2}\). 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 the whole space \(\textbf{R}^{2}\), it must be a shrinking circle. Otherwise the solution must be defined in a strip region.
timetable:
Tue 6 Nov, 17:50 - 18:20, Aula Dini
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