Abstract
AbstractWe examine the L2-gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.
Highlights
We examine the L -gradient ow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16
The convergence results follow from a constrained sharp Reilly-type inequality
We have shown that the length along the elastic ow remains bounded, provided that the initial datum γ ∈ C∞(S, H ) has small elastic energy, more precisely E(γ )
Summary
Our object of study – Euler’s elastic energy – measures the bending of a curve in some Riemannian manifold. For the proof of Proposition 2.13 it is crucial to examine the so-called characteristic integral curves of an elastica These are de ned to be the solutions cz of (2.5), where X = Jγ and z ∈ H is a point of maximum curvature of γ. Let γ be a non-circular elastic curve, i.e. κ is nonconstant It follows from Proposition 2.12 and Lemma 2.10 that for some constants a, b, c ∈ R it holds (κ − λ) γ + κ −γ = a γ − γ + b γ + c . In the case of a circular elastica, i.e. κ ≡ const we obtain that γ is a circle in H since according to Proposition 2.15 this is the only closed curve with constant curvature.
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