The Cauchy–Riemann strain functional for Legendrian curves in the 3-sphere

Abstract

The lower-order cr-invariant variational problem for Legendrian curves in the 3-sphere is studied and its Euler-Lagrange equations are deduced. Closed critical curves are investigated. Closed critical curves with non-constant cr-curvature are characterized. We prove that their cr-equivalence classes are in one-to-one correspondence with the rational points of a connected planar domain. A procedure to explicitly build all such curves is described. In addition, a geometrical interpretation of the rational parameters in terms of three phenomenological invariants is given. (4.2) 4.2. The strain functional and its critical curves. Let J ⊂ I be a closed interval. The integral S J (γ) = J sdt,

The Constrained Blaschke Functional for Spherical Curves

“…A similar relation between elastic curves and the nonlinear Schrödinger equation can be obtained applying the Hasimoto transformation ( [31,32,37]). This phenomenon occurs in other contexts such as Lorentzian, centro-affine, equi-affine, projective and pseudo-conformal geometries ( [12,13,14,19,20,45,47,48,49,55,60]).…”

Integrable geometric flows for curves in pseudoconformal S3

“…□ Remark 4. Proposition 5 is an adaptation to the context of pseudo-Hermitian geometry of the classification of Legendrian curves with constant CR-curvature given in [24]. It follows from Propositions 4 and 5 that the Maslov index of γ m,n is m − n. Computing the writhe of the Lagrangian projection of p H • γ m,n , we find that the Bennequin invariant of γ m,n is −mn.…”

“…Recall from (6) that γ xx = −β 2 γ + b γ x where b := β −1 β x + iβk. Next, we differentiate the expansion ( 17) twice, substitute into (20), and use the inner product formulas…”

“…We substitute this in (24), use dx = β −1 ds to apply integration by parts with respect to s, and substitute expressions for k from (18) and for β from ( 21), giving…”

“…In previous work [4,20,23,24] we investigated geometric invariants and geometric evolution equations for Legendrian curves as well as curves transverse to the contact distribution in the pseudoconformal 3-sphere. The setting of this article is pseudo-hermitian CR geometry, a sub-geometry of CR geometry in which a contact form is specified, resulting in a compatible hermitian metric on the contact planes.…”