Ancient solutions for flow by powers of the curvature in $\mathbb R^2$

Abstract

We construct a new compact convex embedded ancient solution of the κ α flow in R 2 , α ∈ ( 1 2 , 1) that lies between two parallel lines. Using this solution we classify all convex ancient solutions of the κ α flow in R 2 , for α ∈ ( 2 3 , 1). Moreover, we show that any non-compact convex embedded ancient solution of the κ α flow in R 2 , α ∈ ( 1 2 , 1) must be a translating solution.

Ancient finite entropy flows by powers of curvature in $\mathbb{R}^2$

“…Clearly, self-shrinking ancient solutions has finite entropy, since the entropy does not change under homothetic transformation. However, every nonhomothetic ancient α-CSF discovered in previous researches including [4] and [9] do not have finite entropy. See also [17] for a higher dimensional analogue.…”

“…See also [17] for a higher dimensional analogue. Indeed, the entropy of every non-homothetic ancient α-CSF with α ∈ ( 2 3 , 1] must diverge by [22] and [9]. In this paper, we present families of non-homothetic closed ancient α-CSFs which converge to a self-shrinker 3 as t → −∞ after rescaling.…”

Rotationally symmetric translating solutions to extrinsic geometric flows

“…An important class of singularity models for extrinsic geometric flows are the translating solutions, so named because they evolve by ambient translation with constant velocity. Translating solutions arise in the analysis of singularities directly, as blow-up limits [4,15], and also indirectly, in the sense that convex ancient solutions tend to decompose into configurations of asymptotic translators [5,6,8,9,10,11]. In some cases, it can be shown that translating blow-up limits are necessarily rotationally symmetric [7,16].…”

Classification of ancient flows by sub-affine-critical powers of curvature in $\mathbb{R}^2$

“…In the super-affine-critical case Daskalopoulos-Hamilton-Sesum showed in [26] that a convex closed ancient CSF (α = 1) must be a shrinking circle or an Angenent oval. This result was extended for α ∈ ( 2 3 , 1) in [12] by Bourni-Clutterbuck-Nguyen-Stancu-Wei-Wheeler. On the other hand, closed convex ancient affine normal flows must be shrinking ellipses by Corollary 2.7.…”

“…Theorem 1.6 says that there exist infinitely many closed ancient α-CSFs with α < 1 3 developing Type II singularities. This can be compared with the existence of closed strictly convex smooth Type II ancient α-CSFs with α ∈ ( 1 2 , 1] in [8] and [12]. The Type II ancient flows in [8] and [12] are asymptotic to two parallel lines, and they converges to translators at their ends.…”