On the Type IIb solutions to mean curvature flow

TL;DR

This paper investigates the behavior of Type IIb solutions to mean curvature flow, establishing conditions under which solutions are Type IIb and analyzing their asymptotic limits, which are shown to be translating solitons.

Contribution

It characterizes conditions leading to Type IIb solutions in mean curvature flow and describes their asymptotic behavior as translating solitons.

Findings

Convex entire graphs with specific growth conditions produce Type IIb solutions.

Rescaled limits of mean-convex Type IIb flows under noncollapsing conditions are translating solitons.

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 $\mathbb{R}^n$, $n\geq 2$, satisfying there exist positive constants $\epsilon$, $c$ and $N$ such that $ u'(r)\geq c r^{\epsilon} $ for $r\geq N$, 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.