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Session S38 - Geometric Potential Analysis

Thursday, July 15, 19:35 ~ 20:05 UTC-3

On Ahlfors' Schwarzian derivatives for curves

Martin Chuaqui

Pontificia Universidad Católica , Chile   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

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 the respect to the Möbius group in $\mathbb{R}^n$. We will describe some of the applications of the real Schwarzian to the study of simple curves in $\mathbb{R}^n$, to knots in $\mathbb{R}^3$, 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 $\mathbb{R}^3$, 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.

Joint work with Julian Gevirtz, Brad Osgood (Stanford University), and (the late) Peter Duren.

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