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

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 investi- gated. 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 phe- nomenological invariants is given.