On the Type IIb solutions to mean curvature flow

Abstract

In this paper we study the Type IIb mean curvature flow. We first prove that if the convex entire graph $(y,u(|y|))$ over ${\mathbb{R}}^n$, $n\ge 2$, satisfying there exist positive constants $\epsilon$, $c$ and $N$ such that $u^{\prime }(r)\ge c{r}^{\epsilon }$ for $r\ge N$, the longtime solution to mean curvature flow with initial data $(y,u(|y|))$ must be Type IIb. We also study the asymptotic behavior of Type IIb mean curvature flow and show that the limit of suitable rescaling sequence for mean-convex Type IIb mean curvature flow satisfying $\delta$-Andrews' noncollapsing condition is translating soliton.

to appear in Journal of Differential Equations