Differentiability conditions and bounds on singular points

Gary Roy Spoar

Vol. 61 (1975), No. 1, 289–294

DOI: 10.2140/pjm.1975.61.289

Abstract

It is well-known that a normal arc 𝒜₄ of cyclic order four in the conformal plane contains at most finitely many singular points and in fact at most eleven. This bound can be reduced to four in the case of a strongly differentiable 𝒜₄. Using a characterization of singular points on such arcs this paper shows that strong differentiability is not a necessary condition for this bound. In fact a much weaker condition, viz., the existence of tangent circles, is sufficient to obtain four as the least upper bound.

Mathematical Subject Classification 2000

Primary: 53C75

Milestones

Received: 30 January 1975

Published: 1 November 1975