Communications in Analysis and Geometry

Volume 25 (2017)

Number 1

Quantization of the conformal arclength functional on space curves

Pages: 209 – 242



Emilio Musso (Dipartimento di Scienze Matematiche, Politecnico di Torino, Italy)

Lorenzo Nicolodi (Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Italy)


By a conformal string in Euclidean space is meant a closed critical curve with non-constant conformal curvatures of the conformal arclength functional. We prove that (1) the set of conformal classes of conformal strings is in $\textrm{1-1}$ correspondence with the rational points of the complex domain $\lbrace q \in \mathbb{C} : 1/2 \: \mathrm{Re} \: q \lt 1 / \sqrt{2} , \mathrm{Im} \: q \gt 0 , \lvert q \rvert \lt 1/ \sqrt{2} \rbrace $ and (2) any conformal class has a model conformal string, called symmetrical configuration, which is determined by three phenomenological invariants: the order of its symmetry group and its linking numbers with the two conformal circles representing the rotational axes of the symmetry group. This amounts to the quantization of closed trajectories of the contact dynamical system associated to the conformal arclength functional via Griffiths’ formalism of the calculus of variations.


Möbius geometry of curves, closed trajectories, conformal arclength functional, conformal strings, quantization of trajectories, Griffiths’ formalism, linking numbers

2010 Mathematics Subject Classification

53A04, 53A30, 53A55, 53D20, 58-04

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Authors partially supported by PRIN 2010-2011 “Varietà reali e complesse: geometria, topologia e analisi armonica”; FIRB 2008 “Geometria Differenziale Complessa e Dinamica Olomorfa”; and the GNSAGA of INDAM.

Paper received on 2 August 2014.