The “header image” for this blog is an example of an interesting construction in 2-dimensional conformal geometry, due to Richard Kenyon, that I learned of some time ago; I thought it might be fun to try to explain where it comes from.

The example comes from the idea of a Riemann surface lamination. This is an object that geometrizes some ideas in 1-dimensional complex analysis. The basic idea is simple: given a noncompact infinite Riemannian $2$-manifold $\Sigma$, one gives it a new topology by declaring that two points on the surface are “close” in the new topology if there are balls of big radius in the surface centered at the two points which are “almost isometric”. Points that were close in the old topology are close in the new topology, but points that might have been far away in the old topology can become close in the new. For example, if $\Sigma$ is a covering space of some other Riemannian surface $S$, then points in the orbit of the deck group are “infinitely close” in the new topology. This means that the resulting topological space is not Hausdorff; one “Hausdorffifies” by identifying pairs of points that are not contained in disjoint open sets, and the quotient recovers the surface $S$ (assuming that the metric on $S$ is sufficiently generic; otherwise, it recovers $S$ modulo its group of isometries). Morally what one is doing is mapping $\Sigma$ into the space $\mathcal{M}$ of pointed locally compact metric spaces (which is itself a locally compact topological space), and giving it the subspace topology. In more detail, a point in $\mathcal{M}$ is a pair $(X,p)$ where $X$ is a locally compact metric space, and $p \in X$ is a point. A sequence $X_i,p_i$ converges to $X,p$ if there are metric balls $B_i$ around $p_i$ of diameter going to infinity, metric balls $D_i$ around $p$ also of diameter going to infinity, and isometric inclusions of $B_i,D_i$ into metric spaces $Z_i$ in such a way that the Hausdorff distance between the images of $B_i$ and $D_i$ in $Z_i$ goes to zero as $i \to \infty$. Any locally compact metric space $Y$ has a tautological map to $\mathcal{M}$, where each point $y \in Y$ is sent to the point $(Y,y) \in \mathcal{M}$. Gromov showed (see section 6 of this paper) that the space $\mathcal{M}$ itself is locally compact; in fact, this follows in an obvious way from the Arzela-Ascoli theorem.

If $\Sigma$ has bounded geometry — i.e. if the injectivity radius is uniformly bounded below, and the curvature is bounded above and below — then the image of $\Sigma$ in $\mathcal{M}$ is precompact, and its closure is a compact metric space $\mathcal{L}$. The path components of $\mathcal{L}$ are exactly the Riemann surfaces which are arbitrarily well approximated (in the metric sense) on every compact subset by compact subsets of $\Sigma$. If you were wandering around on such a component $\Sigma'$, and you wandered over a compact region, and were only able to measure the geometry up to some (arbitrarily fine) definite precision, you could never rule out the possibility that you were actually wandering around on $\Sigma$. Topologically, $\mathcal{L}$ is a Riemann surface lamination; i.e. a locally compact topological space covered by open charts of the form $U \times X$ where $U$ is an open two-dimensional disk, where $X$ is totally disconnected, and where the transition between charts preserves the decomposition into pieces $U \times \text{point}$, and is smooth (in fact, preserves the Riemann surface structure) on the $U \times \text{point}$ slices, in the overlaps. The unions of “surface” slices — i.e. the path components of $\mathcal{L}$ — piece together to make the leaves of the lamination, which are (complete) Riemann surfaces. In our case, the leaves have Riemannian metrics, which vary continuously in the direction transverse to the leaves. (Surface) laminations occur in other areas of mathematics, for example as inverse limits of sequences of finite covers of a fixed compact surface, or as objects obtained by inductively splitting open sheets in a branched surface (the latter can easily occur as attractors of certain kinds of partially hyperbolic dynamical systems). One well-known example is sometimes called the (punctured) solenoid; its Teichmüller theory is studied by Penner and Šarić  (question: does anyone know how to do a “\acute c” in wordpress? update 11/6: thanks Ian for the unicode hint).

A lamination is said to be minimal if every leaf is dense. In our context this means that for every compact region $K$ in $\Sigma$ and every $\epsilon>0$ there is a $T$ so that every ball in $\Sigma$ of radius $T$ contains a subset $K'$ which is $\epsilon$-close to $K$ in the Gromov-Hausdorff metric. In other words, every “local feature” of $\Sigma$ that appears somewhere, appears with definite density to within any desired degree of accuracy. Consequently, such features will “almost” appear, with the same definite density, in every other leaf $\Sigma'$ of $\mathcal{L}$, and therefore $\Sigma$ is in the closure of each $\Sigma'$. Since $\mathcal{L}$ is (in) the closure of $\Sigma$, this implies that every leaf is dense, as claimed.

In a Riemann surface lamination, the conformal type of every leaf is well-defined. If some leaf is elliptic, then necessarily that leaf is a sphere. So if the lamination is minimal, it is equal to a single closed surface. If every leaf is hyperbolic, then each leaf admits a unique hyperbolic metric in its conformal class (i.e. each leaf can be uniformized), and Candel showed that this family of hyperbolic metrics varies continuously in $\mathcal{L}$. Étienne Ghys asked whether there is an example of a minimal Riemann surface lamination in which some leaves are conformally parabolic, and others are conformally hyperbolic. It turns out that the answer to this question is yes; Richard Kenyon found an example, which I will now describe.

The lamination in question has exactly one hyperbolic leaf, which is topologically a $4$-times punctured sphere. Every other leaf is an infinite cylinder — i.e. it is conformally the punctured plane $\mathbb{C}^*$. Since the lamination is minimal, to describe the lamination, one just needs to describe one leaf. This leaf will be obtained as the boundary of a thickened neighborhood of an infinite planar graph, which is defined inductively, as follows.

Let $T_1$ be the planar “Greek cross” as in the following figure:

Inductively, if we have defined $T_n$, define $T_{n+1}$ by attaching four copies of $T_n$ to the extremities of $T_1$. The first few examples $T_1,\cdots,T_4$ are illustrated in the following figure:

The limit $T_\infty$ is a planar tree with exactly four ends; the boundary of a thickened tubular neighborhood is conformally equivalent to a sphere with four points removed, which is hyperbolic. Every unbounded sequence of points $p_i$ in $T_\infty$ has a subsequence which escapes out one of the ends. Hence every other leaf in the lamination $\mathcal{L}$ this defines has exactly two ends, and is conformally equivalent to a punctured plane, which is parabolic.

The header image is a very similar construction in $3$-dimensional space, where the initial seed has six legs along the coordinate axes instead of four; some (quite large) approximation was then rendered in povray.

When I was in graduate school, I was very interested in the (complex) geometry of Riemann surface laminations, and wanted to understand their deformation theory, perhaps with the aim of using structures like taut foliations and essential laminations to hyperbolize $3$-manifolds, as an intermediate step in an approach to the geometrization conjecture (now a theorem of Perelman). I know that at one point Sullivan was quite interested in such objects, as a tool in the study of Julia sets of rational functions. I have the impression that they are not studied so much these days, but I would be happy to be corrected.