Gravitational lensing by Elliptical Galaxies, and the Schwarz Function

Abstract

We discuss gravitational lensing by elliptical galaxies with some particular mass distributions. Using simple techniques from the theory of quadrature domains and the Schwarz function (cf. \cite{Sh}) we show that when the mass density is constant on confocal ellipses, the total number of lensed images of a point source cannot exceed 5 (4 bright images and 1 dim image). Also, using the Dive--Nikliborc converse of the celebrated Newton's theorem concerning the potentials of ellipsoids, we show that ``Einstein rings'' must always be either circles (in the absence of a tidal shear), or ellipses.

16 pages, 7 figures based on a pleanry talk at the international conference on analysis and mathematical physics "New Trends in Complex and Harmonic Analysis", May 7 - 12, 2007, Voss, Norway