Elastic curves and Willmore Hopf-Tori

Whitham deformation The moduli space of closed wavelike elastic curves in S 2 .Constrained elastica on S 2 arise from anti-twisted polynomialKilling fields of degree 3:( )iA BX =−B ∗ −iAwhereA = 1 2 (m 2 − κ 2 )λ + λ 3B = −(κ ′′ + 1 2 κ3 − 1 2 m 2κ) + iκ ′ λ + κλ 2 .The branchpoints of the spectral curve are the zeros ofwhere the curvature κ satisfiesdet X = m 6 + m 4 λ 2 + m 2 λ 4 + λ 6κ ′ 2 +14 κ4 − 1 2 m 2κ 2 + 2 √ m 6 κ + 1 4 m2 2 − m 4 = 0 .