Elastic curves and Willmore Hopf-Tori

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A brief history◮ 1691 J. Bernoulli: first characterization of planar elastica◮ 1742 D. Bernoulli: proposed variational techniques◮ 1744 Euler: characterization using variational methods◮ 1984 Langer-Singer: classified elastica on 2-sphere◮ 1985 Pinkall: connection between elastica and Willmore tori◮ 1991 Goldstein-Petrich: relationship to mKdV◮ 2003/04 Arroyo-Garay-Mencia: closed elastica andgeneralizations◮ 2008 Bohle-Peters-Pinkall: constrained elastica, andconstrained Willmore Hopf-tori◮ 2012 L.Heller, J.Zentgraf: PhD theses on constrainedWillmore Hopf-tori etc.
Outline of talk:◮ Joint work with F. Pedit and N. Schmitt◮ Finite type curves on S 2◮ Polynomial Killing fields and spectral curves◮ Whitham deformation◮ Constrained elastic curves◮ Moduli space of constrained Willmore Hopf-tori
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 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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