Combinatorial harmonic coordinates
Uniformizing combinatorial annuli.
Can a combinatorial structure determine a rigid geometry ? Here are interesting cases where this works.
Theorem 1 (Thurston, Rodin-Sullivan, Schramm-He, Beardon-Stephenson, Colin de Verdière…) Cover a planar domain with small equal circles. Apply Koebe’s theorem, get a piecewise affine map of the unit disk into mapping centers of circles to centers of circles. As size of circles tends to zero, this map converges uniformly to a Riemann mapping. The ratio of radii of corresponding circles converges to the modulus of the derivative of the Riemann mapping.
Our work : We construct flat surfaces starting from combinatorial data. This can be viewed as a discrete uniformization, in the spirit of Schramm and Cannon-Floyd-Parry.
2. Boundary values on graphs
Let be -connected. Split its boundary into wher is the outermost component. Triangulate it. Let a symmetric conductance function. Then Laplacian makes sense,
Harmonic functions satisfy at inner vertices.
The discrete Dirichlet boundary value problem (D-BVP) consists in finding a harmonic function with prescribed boundary values constant at and at .
The solution is used in the following theorem, in specifying the target space (replacement for the unit disk).
3. A warm up
Theorem 2 (Brooks-Smith-Stone-Tutte 1940) Let be an annulus, a positive number. Let be the straight Euclidean cylinder with height and circumference
Then there exists a mapping which associates to each edge of a unique embedded Euclidean rectangle in in such a way that the collection of these rectangles form a tiling of .
This map preserves energy. It seems that Dehn already had the idea of using Kirckhhoff’s laws in 1903, and pointed out difficulties which are still. This was clarified by Cannon-Floyd-Parry (1994) and Benjamini-Schramm (1996). I have an alternate proof.
The difficulty is that the mapping cannot be extended to a homeomorphism
We shall make a change of charts: We view the given triangulaton as a set of initial charts, and we shall improve on it.
4. A new theorem
Theorem 3 Let be the concentric Euclidean annulus with inner and outer radii and period of (see below).
Then there exist a cellular decomposition of and
- a tiling of by annular shells,
- a homeomorphism mapping each quadrilateral in onto a single annular shell in , and preserving area.
4.1. What goes into the proof
Fact: the level curves of foliate . No critical points.
4.2. Construction of a combinatorial angle
We define a new function, , on , on the annulus minus a slit, the conjugate function of . It is obtained by summing the normal derivatives of along a suitably chosen PL path, joigning the slit to a vertex.
Properties: Level curves of have no endpoints in the interior, and join to . Any two are disjoint. The intersection number between level curves of and level curves of is 1.
4.3. Constructing a rectangular net
Consider the collection of level sets of containing all vertices of . So and . Do the same for .
Definition 4 A rectangular combinatorial net on i a cellular decomposition of where each 2-cell is a simple quadrilateral, and a pair of functions and which satisfy
for all edges.
Theorem 5 There exists a choice of conductances such that and and their level sets form a rectangular combinatorial net.
5. Higher connectivity
Most of the discussion extends to higher connectivity domains.
Split along singular level sets of . Components need not be annuli (this makes it hard). These level sets have a naturally defined length, in terms of the period of a conjugate function . This allows to pile up model annuli and get a model surface of high connectivity.