The String Coffee Table

At that workshop, Prof. Bödigheimer made me tell him about our construction of the String group in terms of the nerve of a 2-group and how that gives rise to a notion of surface transport in $\mathrm{String}(n)$-bundles (or rather in a corresponding 2-bundle with connection).

In that approach, as in the Stolz/Teichner conception of elliptic objects, you want to be able to compose the (Riemann) surface over which the transport takes place out of many topologically disk-shaped little pieces, using composition and identification of boundaries. For bare surfaces this is more or less straightforward, for conformal surfaces, however, this is subtle.

Stolz/Teichner suggest to handle this issue by equipping every little piece of conformal surface suitably with “collars” along which gluing can be performed. While straightforward, the resulting construction might seem a little intricate.

Prof. Bödigheimer pointed out a result he has been working on over the years, which may suggest a rather elegant way to handle this issue, one that should seamlessly lend itself to incorporation in 2-functorial transport over Riemann surfaces.

The idea is to find something like a “normal form” of a representative for an equivalence class of Riemann surfaces obtained by piecing together trivial (“uniform”) surfaces using appropriate combinatorics.

The approach goes back to

D. Hilbert
Zur Theorie der konformen Abbildungen
Nachr. Königl. Ges. Wiss. Göttingen,
math.-phys. Klasse, 314-323 (1909)

and is hence called Hilbert Uniformization.

Bödigheimer’s contribution to this approach is summarized in the ultra-short letter

C.-F. Bödigheimer
Hilbert uniformaization of Riemann surfaces: I
(2004)

A more detailed description can be found in the preprint

C.-F. Bödigheimer
Moduli Spaces of Riemann Surfaces with Boundary
(2003).

An improved version of this has been announed to appear within weeks. Check this website for more. On that site one in particular finds a series of files containing a scanned-in version of

C.-F. Bödigheimer
On the topology of moduli spaces, Part I: Hilbert uniformization
Math. Göttingensis
Heft 7+8 (1990).

(I haven’t seen this one, though. For some strange reason my current equipment does not allow me to view these files.)

In this approach, one builds all Riemann surfaces (possibly with some restriction on the number of boundaries etc.) by choosing a couple of annuli, cutting radial slits into them and identifying left and right sides of these slits among each other.

Alternativley, one can think of the wedges obtained by slicing these annuli, after a conformal transformation, as a collection of standard rectangles in the complex plane, with certain combinatorial prescription for how to glue them. From this point of view the “normal form” for Riemann surfaces becomes particularly nice, being more or less just an $n \times m$ array of standard conformal rectangles with combinatoric data describing their composition along horizontal and vertical boundaries. No auxiliary collars are needed, gluing is exactly along the rectangle’s boundaries.

Every Riemann manifold with at least one (marked) ingoing and at least one outgoing boundary circle can be represented this way.

There is a particularly nice procedure for decomposing any given conformal surface (with at least two boundary circles) canonically into a collection of conformal rectangles with combinatoric identifications (section 6 in the above linked document):

On every such surface, there is precisely one real-valued function $\Phi$ which

$\bullet$ is harmonic

$\bullet$ has no singularities or critical points on the boundary

$\bullet$ takes the value $0$ on all boundary curves labelled “incoming”

$\bullet$ takes the value $1$ on all boundary curves labelled “outgoing”.

Using any representative of the conformal metric on the surface, we can form the gradient of that function and consider the flow induced by that gradient. Flow lines will

$\bullet$ start either at an incoming boundary or at at critical point of $\Phi$

and

$\bullet$ end either at an outgoing boundary or at at critical point of $\Phi$.

The critical points of $\Phi$ together with the set of flow lines which start or end at any critial point form a graph, and this graph, one can show, partitions the Riemann surface into regions which are conformally equivalent to rectangles $[0,1]\times [a,b] \subset \mathbb{C}$.

There are a couple of results concerning the nature of the space of these “normal forms” for Riemann surfaces by students of Bödigheimer. For instance the recent

Johannes F. Ebert, Roland M. Friedrich
The Hilbert-uniformization is real-analytic
math.DG/0601378 .