Quadratic
differentials, Loewner chains and
evolving Fuchsian groups
Jonathan Tsai
The
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 halfplane and G
is a Fuchsian group fixing H. Now, if we deform R by
cutting along a curve g
: (0, T] _{} R that starts from the boundary, then R_{t}_{ }= R \ g (0, t] is conformally
equivalent to H/G_{t} for some Fuchsian group G_{t}. In this seminar, I will introduce a
system of differential equations which describes how the family of Fuchsian groups, (G_{t}), 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 
