The Conformal Arclength Functional

Abstract

A generic smooth curve of the three dimensional Mobius space admits a natural parameter (or conformal arclength). Integrating the conformal arclength we get a conformally invariant variational problem. In the present paper we study the extremal curves of this variational problem. We derive the associated Euler-Lagrange equations and we get the natural equations of the extremal curves. The natural equations are integrated and the explicit solutions are given. IntroductionIn 1980 C. SCHIEMANGK-R. SULANKE ([lo]) and in 1981 R. SULANKE ([ll]) considered the geometry of curves and surfaces in n-dimensional spheres with respect to the action of the group of all orientation-preserving conformal transformations.Section 3 of [lo] and all the paper [ll] are devoted to the geometry of curves in Mobius spaces (mainly in the 3-dimensional case). For generic curves a natural parametrization is founded, a natural lift into the corresponding conformal group is given (the conformal Frenet frame field of the curve) and generalized Frenet formulas are obtained.If we take a curve g(s) of lR3 parametrized by the euclidean arclength s, then the conformal parameter c is given by dc = (k" + k2z2)z ds, where k and z are the curvature and the torsion respectively and k' is the derivative of k with respect to the euclidean arclength s. If d{I, # 0 for all s then the curve is called generic.For generic curves we have that t; is a conformally invariant parametrization and we call the parameter ( the conformal arclength. There are two conformal curvatures, say k , and k , which together with t; give complete information of the geometry of the curve (up to conformal transformations of the ambient space).It seems that the conformal arclength and the conformal curvatures for curves in the 3-dimensional Mobius space were discovered earlier. In [lo] and [ll] the following references are given: H. LIEBMANN, 1923 [8], E. VESSIOT, 1925 [13] and T. TAKASU, 1928 [12].

On the geometry of curves and conformal geodesics in the Möbius space

“…Using Griffiths' formalism (see [14]), in [20] the author wrote the Euler-Lagrange equations for n = 3, obtaining the following system of ODEs for the conformal curvatures μ 1 , μ 2 ∈ C ∞ (I ):…”

Closed trajectories of the conformal arclength functional

“…The critical curves with periodic curvature functions are characterized by the Euler-Lagrange equations [12] …”

“…Using the conservation laws it can be checked (see [12]) that, up to conformal transformations, the parametrization of a critical curve with natural parameters (a, b) ∈ Σ is given by…”

“…In more recent times, the topic has been considered in connection with the regularization of the Kepler problem [6,9,14], within the theory of integrable systems and in the topology of knots [3,1,2,5,8,10,11]. In a previous paper, published several years ago [12], I studied the variational problem defined by the conformal density of a space curve, improperly called conformal arc-element. In that paper I computed the critical curves in terms of elliptic integrals.…”

Coisotropic variational problems

“…We refer the reader to [12] for the explicit construction of the Frenet system of generic curves in the affine space R 3 , to [7,23] for the Frenet system of generic curves in RP 2 , to [20,26] for the Frenet system of generic curves in the conformal 3-sphere and to [22] for the Frenet systems of generic Legendrian curves in the strongly pseudoconvex real hyperquadric Q 3 of CP 2 .…”