Schwarz Reflection Geometry

"The Schwarz function of an analytic arc in the complex plane was introduced by Davis and Pollak in the 1950’s and extensively developed and generalized as a basic analytical tool in monographs by Davis (1974) and Shapiro (1992). Applications of the Schwarz function, since its initial introduction, have appeared occasionally in the literature—e.g., in a series of papers by Gustafsson on quadrature domains and in recent work of Wiegmann, Zabrodin and others on completely integrable planar models. My talk will begin with a brief introduction to the notion of Schwarz function and a review of some of its basic properties and applications. Then I will describe the role of Schwarz functions in Schwarz reflection geometry. Namely, Schwarzian reflection of one analytic plane curve in another may be used to define a formal symmetric space structure on the infinite dimensional space of such curves (joint work with A. Calini). We introduce a computational formalism which takes advantage of the remarkable fit between the Schwarz function and Ottmar Loos’s abstract approach to symmetric spaces. We discuss the resulting geodesic equation, a second order partial differential equation for a time-dependent Schwarz function S(t,z), and describe solutions in terms of meromorphic differentials. Data associated with poles and periods of such differentials will be seen to govern the conformal geometric features of the corresponding singular foliations. The above ideas and constructions will be explained with the help of concrete graphical examples, starting with familiar circle patterns of potential theory—interpreted as geodesics in a three-dimensional Lorentzian manifold of circles. Further, we describe a local normal form for geodesics in M obtained by examining the local action of the group G of analytic circle diffeomorphisms