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.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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