Weierstrass points and Delauney decompositions
by Iain Aitchison
Abstract: We give an overview of some results concerning deformations of geometric structures based on Delauney decompositions and circle packings on hyperbolic Riemann surfaces.
For surfaces with marked points, we reprove a theorem of Nakanishi and Naatanen relating the Teichmuller spaces of closed and punctured surfaces via cone metrics. Weierstrass points on a Riemann surface are invariant under hyperbolic isometry, and we describe how these can be used to define an equivariant cell structure for the Teichmuller space of closed surfaces with respect to the mapping class group. This uses ideas developed by Rivin, Leibon and Springborn. Such a structure has been sought by Ed Witten and Dennis Sullivan, who suggested (December 2006) that Weierstrass points might be useful.
This talk should be viewed as background to Armando Rodado's recent work finding an explicit family of 10 six-dimensional polyhedra, copies of which tessellating the Teichmuller space of genus 2 surfaces, based on Weierstrass points, which will be presented in a later talk.
For More Information: Lawrence Reeves email: L.Reeves@ms.unimelb.edu.au