Unique asymptotics of ancient convex mean curvature flow solutions

Abstract

We study the compact noncollapsed ancient convex solutions to Mean Curvature Flow in R n+1 with O(1) × O(n) symmetry. We show they all have unique asymptotics as t → −∞ and we give precise asymptotic description of these solutions. In particular, solutions constructed by White, and Haslhofer and Hershkovits have those asymptotics (in the case of those particular solutions the asymptotics was predicted and formally computed by Angenent [2]).

On the Construction of Closed Nonconvex Nonsoliton Ancient Mean Curvature Flows

“…We now show that the profile curves shrink to a point, which proves the remaining claim of item (2).…”

“…Thus, t i → 0. The convergence to a circle will follow from Lemma 3.5 below, completing the proof of item (2). The control on its radius (item (3)) will then follow from the following lemma.…”

“…An important example for our construction below is the self-shrinking S 1 × S n−1 constructed by 2 Drugan and Nguyen in [17]. It is rotationally symmetric and has entropy less than 2 [6,17] (by entropy here we mean in the sense of Colding and Minicozzi [14]: the supremum of recentered and rescaled Gaussian densities) -this is explained more below in the proof.…”

“…(4) There exists C < ∞ such that, after parabolically rescaling by d i to obtain flows { T i By items (3) and (4), a subsequence of the flows { T i t } t∈[−d 2 i ,t i ) converges in the smooth topology (uniformly on compact time subintervals) to an ancient mean curvature flow {M t } t∈(−∞,t * ) . By (2) and (3), the limit flow is nonempty and its limit set (at time t * ) is a circle of radius 1. Moreover, by (3), the diameter is type I, in the sense of Lemma 3.2 , and thus the limit flow is compact.…”

“…These examples show that nonconvex ancient solutions can be quite flexible. On the other hand, convex ancient solutions appear to be quite rigid: the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals [8,15] and in higher dimensions there are further partial results in this direction [2,3,7,9,10,23,29]. Moreover, an important result of X.-J.…”