ON THE EXTREMAL CURVATURE AND TORSION OF STEREOGRAPHICALLY PROJECTED ANALYTIC CURVES

Abstract

In this paper, we first derive formulas for the curvature and torsion of curves on $S^2$ produced by stereographically projecting the image curves of analytic, univalent functions belonging to the class $\mathcal S$. We are concerned here with the problems of determining the extreme values of the curvatures and torsions, as well as the functions belonging to $\mathcal S$ which attain these extreme values. An analysis of the asymptotic behavior of these curvature and torsion formulas will allow for the formulation of plausible conjectures.