Geometry Seminar

 

Quadratic differentials, Loewner chains and

evolving Fuchsian groups

 

Jonathan Tsai

The Chinese University of Hong Kong

 

 

Abstract

Let R be a Riemann surface with boundary. In most cases R is conformally equivalent to the quotient space H/G where H is the upper half-plane and G is a Fuchsian group fixing H. Now, if we deform R by cutting along a curve g : (0, T] image004 R that starts from the boundary, then Rt = R \ g (0, t] is conformally equivalent to H/Gt for some Fuchsian group Gt.  In this seminar, I will introduce a system of differential equations which describes how the family of Fuchsian groups, (Gt), changes as we cut along the curve g. This can be viewed as a generalization of the Loewner differential equation to Riemann surfaces. We will see that the simplest case is when g is a trajectory arc of a certain quadratic differential. We will also look at how this system of differential equations can be solved numerically.

 

 

Date:

February 20, 2008 (Wednesday)

Time:

3:00 – 4:00pm

Place:

Room 517, Meng Wah Complex, HKU

 

 

 

All are welcome