Everytopic Seminar

October 31st, 2008, 1:40 pm – 3:00pm, Goldsmith 226



Title: Harmonic functions, graphs, and tesselations of Riemann surfaces

Speaker: Carl-Friedrich Boedigheimer, University of Bonn



On a Riemann surface $F$ (with or without boundary)

let a finite number of points $P_i$ with tangent vectors $V_i$ be

specified. Choosing a type $\nu_i = 0, 1, 2, \ldots $ and a

positive real coefficient $A_i$ for each point,

this determines a unique real-valued harmonic function

$u : F \to \mathbb{R} \cup \infty$.

The gradient flow of $u$ gives a graph $\mathcal{K}$, with the above

points $P_i$ and the critical points as vertices and flow lines

and level curves as edges.

The complement of that graph is a tesselation of the surface $F$.

(Using that $u$ and the graph varies continuously with the complex

structure, this gives a cell decomposition of the moduli space.)