Elastic curves and Willmore Hopf-Tori

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Whitham deformationExampleThe branchpoints of the spectral curve of an elastic curve are theroots of the real even quartic polynomialm 4 + m 2 λ 2 + λ 4 .With discriminant ∆ = 1 4 m 2 − m 4 , away from degeneratesituations, there are two caseswave-like if ∆ < 0: distinct branchpoints ±p 1 , ±p 2 with p 2 = ¯p 1orbit-like if ∆ > 0: distinct branchpoints on iR × .In the case ∆ = 0 the spectral curve is singular.Figure: Spectral curves of wave-like and orbit-like elastic curves.
Whitham deformation Flow of elastic curves in S 2Flowing off a great circle by opening up a pair of branchpointsincreases the spectral genus from g = 0 to g = 1. The flowthrough closed elastic curves of spectral genus g = 1 are given byṗ 1 = (1 − p 2 1)(p 2 + p 1 β) ,ṗ 2 = (1 − p 2 2)(p 1 + p 2 β) .Qualitative analysis of these equations yield a picture of the modulispace of closed elastic curves on S 2 .
Page 2 and 3: A brief history◮ 1691 J. Bernoull
Page 4 and 5: ◮ If h : S 3 → S 2 is the Hopf-
Page 6 and 7: Figure: A deformation through const
Page 8 and 9: Finite type curves in S 2Curves in
Page 10 and 11: Finite type curves in S 2Reconstruc
Page 12 and 13: Polynomial Killing fields and spect
Page 14 and 15: Polynomial Killing fields and spect
Page 16 and 17: Whitham deformationPolynomials A, B
Page 18 and 19: Whitham deformationIncreasing the s
Page 22 and 23: Whitham deformation Flow of elastic
Page 24 and 25: Whitham deformation The moduli spac
Page 26 and 27: Whitham deformation The moduli spac
Page 28 and 29: Moduli space of constrained elastic
Page 30 and 31: Moduli space of constrained elastic
Page 32: Moduli space of constrained elastic

Elastic curves and Willmore Hopf-Tori

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