Geometry Seminar


Quadratic differentials, Loewner chains and

evolving Fuchsian groups


Jonathan Tsai

The Chinese University of Hong Kong




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.




February 20, 2008 (Wednesday)


3:00 – 4:00pm


Room 517, Meng Wah Complex, HKU




All are welcome