Ahlfors’ Schwarzians for curves

Abstract: We discuss Ahlfors' Schwarzian derivatives for curves in euclidean space introduced some three decades ago. The definitions consider separate generalizations of the real and imaginary part of the classical operator in the complex plane that have important invariance properties with respect to the Möbius group in euclidean n-space. We will describe some of the applications of the real Schwarzian to the study of simple curves in n-space, to knots in 3-space, as well as to the injectivity of the conformal parametrization of minimal surfaces in 3-space. The role of the imaginary Schwarzian will be presented in euclidean 3-space, highlighting its connection with the osculating sphere, a new transformation law under the Möbius group, and theorems on the existence and uniqueness of parametrized curves with prescribed real and imaginary Schwarzians.

complex variables

Audience: researchers in the topic

CAvid: Complex Analysis video seminar

Series comments: Please e-mail R.Halburd@ucl.ac.uk for the Zoom link. Also, please let me know whether you would like to be added to the mailing list to automatically receive links for future talks in CAvid.