12
The Spectral Geometry of Riemann Surfaces
R. Brooks
Summary.This chapter is spread out over a number of papers and also builds on
my earlier work on the relationship between the spectral geometry of manifolds and
the spectral geometry of graphs. It seemed to be a reasonable idea to put together
these ideas in one overall framework, which will be accessible to someone at the
graduate level.
The material naturally breaks up into a number of areas, each one having connec-
tions to basic graduate material, but putting these different pieces together demands
a fair amount of breadth. We hope to supply this breadth in the chapters.
12.1 Introduction
These are notes to accompany my lectures at the Institut Henri Poincar ́e. The
idea of these lectures is to present the approach of myself and Eran Makover
toward understanding the spectral geometry of a typical Riemann surface.
This work is spread out over a number of papers and also builds on my
earlier work on the relationship between the spectral geometry of manifolds
and the spectral geometry of graphs. It seemed to be a reasonable idea to
put together these ideas in one overall framework, which will be accessible to
someone at the graduate level.
Unfortunately, this is not the approach that we will take in these notes.
The material naturally breaks up into a number of areas, each one having
connections to basic graduate material, but putting these different pieces to-
gether demands a fair amount of breadth. We hope to supply this breadth in
the lectures.
Our hope in these notes is somewhat more modest. Each section of the
notes will be devoted to a section of the material. Our plan is to make each
section pretty much independent, so that someone can pick up a particular
topic. The task of knitting the different pieces together to get a coherent
overall picture will have to wait, perhaps for a long time. In the meantime, it
is hoped that the various sections will appear in a manner that will allow the
students to keep up with the lectures.