Conformal arc-length as $\frac{1}{2}$ dimensional length of the set of osculating circles

Abstract

The set of osculating circles of a given curve in $\SS^3$ forms a curve in the set of oriented circles in $\SS^3$. We show that its "$\frac{1}{2}$-dimensional measure" with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.

arXiv comment39 pages, 11 figures. To appear in Comm. Math. Helv