Elastic curves and Willmore Hopf-Tori

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Finite type <strong>curves</strong> in S 2Curves in S 2Let γ : R → S 2 be a smooth immersed arc-length parametrizedcurve with geodesic curvature κ. The frame G : R → SO 3 of γwith columnsG = (γ, γ ′ , γ × γ ′ )satisfies⎛0 −1⎞0G ′ = G ⎝1 0 −κ⎠ .0 κ 0Here prime denotes differentiation with respect to arc-length.
Finite type <strong>curves</strong> in S 2mKdV hierarchyThe mKdV hierarchy is an infinite hierarchy of evolution equationswith first three equations˙κ = −κ ′˙κ = −k ′′′ − 3 2 κ2 κ ′˙κ = −k ′′′′′ − 15 8 κ4 κ ′ − 5 2 κ2 κ ′′′ − 10κκ ′ κ ′′.A curve is called of finite type if κ is stationary under all but fintelymany of the flows in this hierarchy.
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 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 20 and 21: Whitham deformationExampleThe branc
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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