Elastic curves and Willmore Hopf-Tori

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Polynomial Killing fields and spectral curveSpectral curveClosing conditionsLet X be a periodic polynomial Killing field with period ρ, andextended frame F λ . Then the corresponding curve is closed if andonly if the monodromy M λ = F λ (ρ) satisfiesM 1 = M −1 = 1 or M 1 = M −1 = −1 .From F −1λX 0F λ = X and X 0 = X(ρ) it follows that[X 0 , M λ ] = 0 .An eigenvalue µ of M λ is a meromorphic function on Σ whose onlypoles are simple poles over the two points ∞ ± .
Whitham deformationPolynomials A, B and CWhitham deformationHence d log µ is an abelian differential of the 2 nd kind on Σ, so ofthe formd log µ = √ B dλAwhere A = det X and λ ↦→ B(λ) is a real polynomial of degreed = g + 1.Suppose A and B depend on an additional parameter t ∈ R. Canwriteddt log µ = √ C Awhere λ ↦→ C(λ) is a polynomial of degree at most g.
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 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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