## Notes of Sa’ar Hersonsky’s lecture

Combinatorial harmonic coordinates

Uniformizing combinatorial annuli.

1. Perspective

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 ${\Omega}$ with small equal circles. Apply Koebe’s theorem, get a piecewise affine map of the unit disk ${D}$ into ${\Omega}$ 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 ${\Omega}$ be ${m}$-connected. Split its boundary into ${E_1\cap E_2}$ wher ${E_1}$ is the outermost component. Triangulate it. Let ${c:T^{(1)}\rightarrow{\mathbb R}_+}$ a symmetric conductance function. Then Laplacian makes sense,

$\displaystyle \begin{array}{rcl} \Delta u (x)=\sum_{x\sim y}c(x,y)(u(x)-u(y)). \end{array}$

Harmonic functions satisfy ${\Delta u=0}$ at inner vertices.

$\displaystyle \begin{array}{rcl} E(u)=\sum_{x\sim y}c(x,y)(u(x)-u(y))^2. \end{array}$

The discrete Dirichlet boundary value problem (D-BVP) consists in finding a harmonic function with prescribed boundary values ${g=k}$ constant at ${E_1}$ and ${g=0}$ at ${E_2}$.

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 ${A}$ be an annulus, ${k}$ a positive number. Let ${S_A}$ be the straight Euclidean cylinder with height ${k}$ and circumference

$\displaystyle \begin{array}{rcl} C=\sum_{x\in E_1}\frac{\partial g}{\partial n}(x). \end{array}$

Then there exists a mapping ${f}$ which associates to each edge of ${A}$ a unique embedded Euclidean rectangle in ${S_A}$ in such a way that the collection of these rectangles form a tiling of ${S_A}$.

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 ${S_A}$ be the concentric Euclidean annulus with inner and outer radii ${r_1=1}$ and ${r_2=2\pi/}$ period of ${\theta}$ (see below).

Then there exist a cellular decomposition ${R}$ of ${A}$ and

1. a tiling ${T}$ of ${S_A}$ by annular shells,
2. a homeomorphism ${f:A\rightarrow S_A}$ mapping each quadrilateral in ${R^{(2)}}$ onto a single annular shell in ${S_A}$, and preserving area.

4.1. What goes into the proof

Fact: the level curves of ${g}$ foliate ${A}$. No critical points.

4.2. Construction of a combinatorial angle

We define a new function, ${\theta}$, on ${T^{(0)}}$, on the annulus minus a slit, the conjugate function of ${g}$. It is obtained by summing the normal derivatives of ${g}$ along a suitably chosen PL path, joigning the slit to a vertex.

Properties: Level curves of ${\theta}$ have no endpoints in the interior, and join ${E_1}$ to ${E_2}$. Any two are disjoint. The intersection number between level curves of ${\theta}$ and level curves of ${g}$ is 1.

4.3. Constructing a rectangular net

Consider the collection ${L=\{L(v_0),\ldots,L(v_k)\}}$ of level sets of ${g}$ containing all vertices of ${T}$. So ${L(v_0)=E_2}$ and ${L(v_k)=E_1}$. Do the same for ${\theta}$.

Definition 4 A rectangular combinatorial net on ${\Omega}$ i a cellular decomposition ${R}$ of ${\Omega}$ where each 2-cell is a simple quadrilateral, and a pair of functions ${\phi}$ and ${\psi}$ which satisfy

$\displaystyle \begin{array}{rcl} d\phi(e)d\psi(e)=0 \end{array}$

for all edges.

Theorem 5 There exists a choice of conductances such that ${g}$ and ${\theta}$ and their level sets form a rectangular combinatorial net.

5. Higher connectivity

Most of the discussion extends to higher connectivity domains.

Split ${\Omega}$ along singular level sets of ${g}$. 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 ${\theta}$. This allows to pile up model annuli and get a model surface of high connectivity.