Closed trajectories of the conformal arclength functional

Abstract

Abstract. The purpose of this report is to give a brief overview of some unpublished results about the geometry of closed critical curves of a conformally invariant functional for space curves.

A conformally invariant variational problem for time-like curves

“…The integral of the conformal arcelement defines the simplest conformal invariant variational problem on the space of generic time-like curves of E 1,n−1 , called the conformal strain functional. This is the Lorentzian counterpart of the conformal arc-length functional in Möbius geometry [21,22,24,26,27,28] and generalizes the homonymous functional for time-like curves in E 1,2 , previously considered in [9]. Proceeding in analogy with [24] and using the method of moving frames we deduce the variational equations satisfied by the stationary curves, referred to as conformal world-lines.…”

“…In Section 2, we collect from the literature few basic facts about conformal Lorentzian geometry [2,14] and we reformulate in the Lorentzian context the classical approach to the conformal geometry of curves in the Möbius space [10,16,25,32,33,34,35]. We define the conformal strain and the conformal lineelement, which is the Lorentzian analogue of the conformal arc element of a curve in S n [21,24,25,26,27,28]. In Section 3, we use the moving frame method to compute the Euler-Lagrange equations of the conformal strain functional (Theorem 2).…”

Quantization of the conformal arclength functional on space curves

“…A general reference for Möbius geometry is [15], to which we refer for an updated list of modern and classical references to the subject (see also [16]). The main results of the paper were previously announced in [27].…”

Conformal geometry of timelike curves in the (1+2)-Einstein universe

“…The first is to describe local and global conformal differential invariants of a timelike curve. The second purpose is to address the question of existence and properties of closed trajectories for the variational problem defined by the conformal strain functional, the Lorentzian analogue of the conformal arclength functional in Möbius geometry [25,30,31,34]. The Lagrangian of the strain functional depends on third-order jets and shares many similarities with the relativistic models for massless or massive particles based on higher-order action functionals, a topic which has been much studied over the past twenty years [15,19,23,32,33,35,36,43].…”

“…In Section 3, we study the conformal geometry of timelike curves in the Einstein universe. We define the infinitesimal conformal strain, which is the Lorentzian analogue of the conformal arc element of a curve in S 3 [24,28,30,31,34], and the notion of a conformal vertex. An explicit description of curves all of whose points are vertices is given in Proposition 1.…”