2021-02-06

We discuss an open problem about parametrising boundaries of semi-uniform domains, a class of domains introduced by Aikawa and Hirata. The question is essentially whether one can generalise the result that curves of bounded turning in the plane are quasi symmetric images of the real line, or they are bi-Lipschitz images if additionally they are Ahlfors regular. We first give a general background of the problem.

## Parametrizing curves in the plane

A common type of problem that occurs in several branches of analysis is to classify which sets are actually just images of some smaller and simpler class of sets under a certain class of functions. This is in part a classification problem that’s interesting in this own right; in some sense such a classification theorem tells us that a strange class of objects are more familiar than we realized. These classification theorems are also sometimes useful: If a set is the image of some nice function, we can use the properties of that function to study the geometry of a set. A familiar example is the Riemann mapping theorem, which says any simply connected planar domain is the image of the unit disc under a conformal mapping, and the implications of this are vast (see [5] for example). As a really basic example, given a differentiable function $f:\mathbb{R}\rightarrow \mathbb{R}$, we can use differentiability to conclude that the graph has tangents at every point, that is, the graph looks more and more like a line as you zoom in at each point. In this example, however, we started with a function and drew conclusions about its image, but in practice perhaps there is a set you want to study but you don’t know a priori that it is the graph or image of some function.

In practice, being the image of a differentiable function is a bit much to hope for. It’s not impossible to come up with sufficient conditions for this to be the case, but they are quite stringent. By contrast, finding Lipschitz parametrizations is easier. For example, the following theorem is not hard to verify:

Theorem: Suppose $\Gamma\subseteq \mathbb{R}^{n}$ is a connected set of finite one dimensional Hausdorff measure. Then there is a surjective 1-Lipschitz map $f:[0,2\mathscr{H}^{1}(\Gamma)]\rightarrow \Gamma$.

The result also holds in metric spaces, but we’ll keep things simple here. So while Lipschitz functions are not as nice as differentiable functions, we see that Lipschitz parametrizations are quite easy to find given some simple conditions (connectivity and finite length in this case). And now you can conclude things like the existence of tangents at almost every point in $\Gamma$ using this parametrizing function.

What about bi-Lipschitz functions, that is, Lipschitz with Lipschitz inverse? There are similar criteria to the above, but we need stronger conditions on the structure and size of our set.

In terms of the structure of the set, in addition to assuming connectivity, we’ll need to assume the connected set does not pinch on itself too much, since these are obstacles to bi-Lipschitz parametrizability (think of the McDonald’s logo as a curve, this cannot be the image of a bi-Lipschitz map). There are a few equivalent ways of phrasing this condition (e.g. Ahlfors' 3-point condition), but in this post I’m going to focus on curves in the plane given by boundaries of domains and the condition will be given by the domain instead of the boundary.

A domain $\Omega\subseteq \mathbb{C}$ is uniform if there is a constant $C>0$ so that any pair of points $x,y\in \Omega$ may be connected by a curve $\gamma\subseteq \Omega$ such that (a) its length is at most $C|x-y|$ and (b) for any $z\in \gamma$, $dist(z,\partial\Omega)\geq \min{\ell(x,z),\ell(y,z)}/C$ where $\ell(a,b)$ denotes the length of the portion of the curve $\gamma$ between $a$ and $b$. Condition (a) is says that the curve is not too much longer than the distance between the points $x$ and $y$ and a domain satisfying this said to be quasiconvex. Condition (b) says that the curve moves away from the boundary as it moves away from either $x$ or $y$, or in other words, the domain does not pinch too tightly around the path between the two points. Such a curve is sometimes called a cigar curve.

The first figure above shows a uniform domain and two poin`ts connected by a curve satisfying conditions (a) and (b), where the region between the two dotted lines (i.e. the cigar) is the region $\bigcup{B(z,\min{|z-x|,|z-y|}/C):z\in \gamma}$ which, by assumption should lie within the domain. The center figure is still uniform, but the constant $C$ in the definition will be much larger since condition (b) is harder to satisfy than the first region (because the region is pinching in the middle). The last domain can be shown to satisfy (b) for some fixed $C$ for any pair of points in the domain, but there is no constant $C$ so that (a) holds for all points, since points on opposite side of the slit will require a larger curve to join them in proportion to their mutual distance as they get closer and closer to the circular part of the boundary.

Without any assumptions on the measure yet, we can already get a parametrization result using quasisymmetric maps.

Theorem: If $\Gamma = \partial\Omega$ where $\Omega\subseteq \mathbb{C}$ is a simply connected uniform domain, then there is a quasisymmetric map $f:\mathbb{C}\rightarrow \mathbb{C}$ so that $\Gamma = f(\mathbb{S})$ if $\Gamma$ is unbounded and $\Gamma = f(\mathbb{R})$ if $\Gamma$ is unbounded.

By a quasisymmetric map we mean a function $f:\mathbb{C}\rightarrow \mathbb{C}$ so that, for some H>0, $|f(x)-f(y)|\leq H|f(x)-f(z)|$ whenever $|x-y|\leq |x-z|$. These maps are not bi-Lipschitz necessarily. An example is the case of the domain $\Omega$ whose boundary is the von Koch curve. This is a uniform domain, and in this case the quasisymmetric map will satisfy $|f(x)-f(y)|\sim |x-y|^{s}$ for $x,y\in\mathbb{S}$ where $s=\frac{\log 4}{\log 3}$. A simple way to think about how quasisymmetric maps distort space is as follows: consider two adjacent intervals I and J in the real line of equal size. A bi-Lipschitz map of the real line will take these two intervals to sets that have comparable size to what they had before (i.e. $diam; f(I) \sim diam ; I$); a quasisymmetric map, however, may shrink or increase the size of these intervals without bound, but the ratios of the sizes of these two intervals is preserved up to constant (i.e. $\frac{diam ; f(I)}{diam ; f(J)}\sim 1$). For more details, see [5, Chapter VII.4].

If we assume now some bounds on the measure of the curve $\Gamma$, then we can improve to bi-Lipschitz parametrizability.

Theorem: Suppose $\Gamma = \partial\Omega$ where $\Omega\subseteq \mathbb{C}$ is a simply connected uniform domain. Assume $\Gamma$ is Ahlfors 1-regular, meaning $\mathscr{H}^{1}(\Gamma\cap B(x,r))\sim r$ for all $x\in \Gamma$, $0<r<diam; \Gamma$. Then there is $f:\mathbb{C}\rightarrow \mathbb{C}$ bi-Lipschitz so that $f(\mathbb{S})=\Gamma$ if $\Gamma$ is bounded and $f(\mathbb{R})=\Gamma$ if $\Gamma$ is unbounded.

This can be obtained from the previous theorem: in the case of $\Gamma$ bounded, let $g:\mathbb{S}\rightarrow \Gamma$ be the quasisymmetric map from the previous theorem, then its arclength parametrization $G$ will be bi-Lipschitz due to the Ahlfors regularity condition, and a theorem of Tukia [8] says such a function can be extended to a bi-Lipschitz map of $\mathbb{C}\rightarrow \mathbb{C}$.

## Semi-uniform domains

There is a kind of domain that, as the name suggests, is slightly weaker than being uniform. A domain $\Omega\subseteq \mathbb{C}$ is semi-uniform if there is a constant $C>0$ so that any pair of points $x\in \Omega$ and $y\in \partial\Omega$ may be connected by a curve $\gamma\subseteq \overline{\Omega}$ of length at most $C|x-y|$ and so that, for any $z\in \gamma$, $dist(z,\partial\Omega)\geq \min{|z-x|,|z-y|}/C$.

The only difference from uniformity is that one of the points in the uniformity condition must be in the boundary. So while uniformity says we may travel between any pair of points in the domain without the domain pinching on us, semi-uniformity says we can only do this when traveling from a point in the domain to a point on the boundary.

Above we show the domain that is $\Omega = \mathbb{C}\backslash ((-\infty,-1],[1,\infty))$ along with some examples of pairs of points $x\in \Omega$ and $y\in\partial \Omega$ and the cigar curves between them. Note that this domain is semi-uniform but not uniform, since while any curve has a short cigar path to the boundary, points above and below the lines cannot be connected by a short cigar path since any such path must pass through the hole in the middle (and so either the length or the width of the cigar will worsen the farther the points are from the hole).

In this second image above we have a bounded semi-uniform domain that isn’t uniform, since points on either side of the slit in the domain can only be connected by a curve much larger than their distance between each other (and the curve gets larger as the points approach the slit).

Semi-uniform domains were introduced by Aikawa and Hirata in [1]. Aikawa and Hirata showed that a John domain in $\mathbb{R}^{n}$ satisfying the capacity density condition–a condition that says the boundary of the domain is nondegenerate in some sense, but could be totally disconnected–has doubling harmonic measure if and only if the domain is semi-uniform. Jerison and Kenig showed in [6] that a simply connected planar domain was doubling if the domain was uniform, so Aikawa and Hirata’s result characterizes this doubling property for a class of domains that aren’t simply connected. (For the exact definition of doubling I’m using, see [1].) As far as I’m aware, the only two papers that have studied these domains are theirs and a paper of mine [2].

## The Question

In [2], I studied semi-uniform domains whose boundaries were uniformly rectifiable. I’ll just state a definition for 1-dimensional sets since that’s the context we’re working in: a Ahlfors 1-regular set $E$ is uniformly rectifiable if, for every $x\in E$ and $0<r<diam; E$, there is a curve $\Gamma_{x,r}$ of length at most $Cr$ containing $E\cap B(x,r)$. This is not the standard definition, and in fact there are MANY equivalent definitions of uniform rectifiability, and the interested reader should take a look at the books [3] and [4].

Question: Characterize all uniformly rectifiable sets $E\subseteq \mathbb{C}$ that are boundaries of semi-uniform domains.

We have seen that if $E$ is also the boundary of a uniform domain, then it is bi-Lipschitz equivalent to a line or circle. The above question seeks to get rid of the topological assumptions, that is, in the absence of a connectivity assumption on $E$ or it being the boundary of a simply connected domain, we just assume it is uniformly rectifiable.

My guess would be that such domains whose closures are all of $\mathbb{C}$ are equivalent up to a bi-Lipschitz map to domains of the form $\mathbb{C}\backslash E$ where $E$ is either a subset of the real line or the unit circle. However, there could be a domain I’m missing. This could be a simpler and more feasible start, otherwise for bounded domains we saw the slit disc from earlier, and we could add more slits to it so that it is still semi-uniform with uniformly rectifiable boundary, so it seems like there are a lot more possibilities.

Why think such a thing is true? In [2] I had to construct an example of a semi-uniform domain with uniformly rectifiable boundary, which I did by first constructing the boundary and then showing that the boundary was bi-Lipschitz equivalent to a subset of the real line, so I was curious if this was true in general.

A paper related to this problem could be [7]. There, MacManus shows that bi-Lipschitz maps of subsets of the unit circle into the plane are extendable to bi-Lipschitz maps of the whole plane, generalizing Tukia’s result. There he needs to carefully piece together the image of what is possibly a totally disconnected subset of the circle into the image of a bi–Lipschitz curve, which is analogous to what the above problem is asking.

## Bibliography

1. H. Aikawa and K. Hirata, Doubling conditions for harmonic measure in John domains, Ann. Inst. Fourier (Grenoble) 58 (2008), 429–445. Link.
2. J. Azzam, Semi-uniform domains and the $A_{\infty}$ property for harmonic measure, arXiv preprint arXiv:1711.03088, to appear in IMRN (2017). Link.
3. G. David and S. W. Semmes, Singular integrals and rectifiable sets in ${\bf R}^n$: Beyond Lipschitz graphs, Astérisque (1991), 152.
4. G. David and S. W. Semmes, Analysis of and on uniformly rectifiable sets, American Mathematical Society, 1993.
5. J. B. Garnett and D. E. Marshall, Harmonic measure, Cambridge University Press, 2008.
6. D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), 80–147. Link.
7. P. MacManus, Bi-Lipschitz extensions in the plane, J. Anal. Math. 66 (1995), 85–115. Link.
8. P. Tukia, The planar Schönflies theorem for Lipschitz maps, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), 49–72.