Classification of ancient finite-entropy curve shortening flows

Abstract

We prove that any ancient smooth embedded finite-entropy curve shortening flow is one of the following: a static line, a shrinking circle, a paper clip, a translating grim reaper, or a graphical ancient trombone. An ancient trombone is an immersed ancient flow, either compact or non-compact, obtained by gluing together $m$ translating grim reaper curves. For each $m$, there exists a $(2m-1)$-parameter family of graphical ancient trombones, up to rigid motions and time shifts as constructed by Angenent-You. In particular, our result implies that any compact ancient smooth embedded finite-entropy flow is convex. Moreover, any non-compact ancient smooth embedded finite-entropy flow is either a static line or a complete graph over a fixed open interval.

Figures (5)

• Figure 1: Finger in strip.
• Figure 2: An impossible sheet in case (A1) with $\theta(s_{\mathrm{infl}})<0$.
• Figure 3: Tips asymptotic to grim reapers.
• Figure 4: Case 1: $a_{i+1} < a_{i-1} < a_{i}$: proper nesting of fingers.
• Figure 5: Case 2: $a_{i-1} \leq a_{i+1} < a_i$: spiral and nesting chains.

Theorems & Definitions (65)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4
• Corollary 1.5
• Theorem 3.1: general sharp asymptotic behavior
• Theorem 3.2
• Corollary 3.3
• Lemma 3.4
• proof