Convex ancient solutions to curve shortening flow
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Calculus of Variations
Convex ancient solutions to curve shortening flow Theodora Bourni1 · Mat Langford1,3 · Giuseppe Tinaglia2 Received: 6 March 2019 / Accepted: 31 May 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We show that the only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals, completing the classification initiated by Daskalopoulos, Hamilton and Šešum and X.-J. Wang. Mathematics Subject Classification 53 · 35
1 Introduction A smooth one-parameter family {t }t∈I of connected, immersed planar curves t ⊂ R2 evolves by curve shortening flow if ∂t γ (θ, t) = κ (θ, t) for each (θ, t) ∈ × I for some smooth family γ : × I → R2 of immersions γ (·, t) : → R2 of t , where κ (·, t) is the curvature vector of γ (·, t). The solution {t }t∈I is called ancient if I contains the interval (∞, t0 ) for some t0 ∈ R, which we may, without loss of generality, take to be zero. We refer to a solution as compact if ∼ = S 1 , convex if the timeslices t each bound convex domains (in which case the immersions γ (·, t) are proper embeddings), locally uniformly convex if the curvature κ is always positive and maximal if it cannot be extended forwards in time. The following families of curves constitute maximal convex ancient solutions to curve shortening flow.
Communicated by N. Trudinger.
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Theodora Bourni [email protected] Mat Langford [email protected]; [email protected] Giuseppe Tinaglia [email protected]
1
Department of Mathematics, University of Tennessee Knoxville, Knoxville, TN 37996-1320, USA
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Department of Mathematics, King’s College London, London, WC2R 2LS, UK
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School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, NSW 2308, Australia 0123456789().: V,-vol
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T. Bourni et al.
– The stationary line {Lt }t∈(−∞,∞) , where Lt := {(x, 0) : x ∈ R}. 1 – The shrinking circle {S√ } . −2t t∈(−∞,0)
– The Grim Reaper {Gt }t∈(−∞,∞) , where Gt := {(x, y) : cos x = et−y } [17]. – The Angenent oval {At }t∈(−∞,0) , where At := {(x, y) : cos x = et cosh y} [3]. We will prove that the aforementioned examples are the only ones possible (modulo spacetime translation, spatial rotation and parabolic rescaling). Theorem 1.1 The only convex ancient solutions to curve shortening flow are the stationary lines, shrinking circles, Grim Reapers and Angenent ovals. Theorem 1.1 completes the classification of convex ancient solutions to curve shortening flow initiated by Daskalopoulos et al. [18]. Indeed, Daskalopouloset al. showed that the shrinking circles and the Angenent ovals are the only compact examples [10]. Their arguments are based on the analysis of a certain Lyapunov functional. On the other hand, Wang’s results imply, in particular, that a convex ancient solution {t }t∈(−∞,0) must either be entire (i.e. sweep out the whole plane, in the sense that ∪t 0. Taking δ → 0 then yields the claim.
The Alexandrov reflection principle
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