Convergence of Curve Shortening Flow to Translating Soliton

Abstract

This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in R 2 under the α-curve shortening flow for exponents α > 1 2 . We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under α-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case α = 1, and we prove for all exponents up to the critical case α > 1 2 .

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

“…Finally, we show that any ancient non-compact convex solution to the κ α flow, for all α ∈ (1/2, 1], must be a translating solution, therefore it is unique modulo rigid motions and parabolic rescalings by [14]. We remark that this in fact is a corollary of the more involved result [7,Theorem 1.1].…”

“…Γ t − γ(e 2 , t) converges, as t → +∞ to a translator T 0 . In the case the solution is between two parallel lines, T 0 = T by [7,Theorem 1.1]. In case the solution is entire, then so is T 0 (which can be easily verified, for example by using translating solutions as barriers) and thus again T = T 0 .…”

“…The convex translating solution. According to the theorem of Urbas stated below [14] (see also [7,Proposition 2.4]), modulo translations, there is a unique convex translating solution of the κ α -flow moving with speed 1 along the e 2 direction, for α ∈ (1/2, 1], other than the straight line. We thus have that the limit Γ s t = Γ t+s − γ(θ, s), as s → −∞, exists.…”

“…T is either a straight line or it is defined between two parallel lines [14]. B. Choi, K. Choi and Daskalopoulos in [7] observed that, by the result in [9], a convex complete graph over an open interval I (either bounded or unbounded), under the flow, remains a convex complete graph over the same interval I. Hence, the solution is either entire or lies between two parallel lines.…”

Convergence of Gauss curvature flows to translating solitons

“…Recently, the authors [12] showed the convergence to a translating soliton for n = 1 and α > 1 2 . In this paper, we establish its higher dimensional result for n ≥ 2 as follows.…”