10 equivalent ways of defining quasicircles

I have been dealing with a lot of quasicircles lately, and I was thinking it would be practical to write a post about its many equivalent definitions. Quasicircles are Jordan curves which satisfy certain nice analytic and geometric conditions. While conformal maps of the Riemann sphere (aka Möbius transformations) map circles to circles, quasiconformal maps map quasicircles to quasicircles.

We will denote by $\hat{\mathbb{C}}$ the Riemann sphere $\mathbb{C} \cup \{ \infty \}$. Denote by $\mathbb{D}$ the standard unit disk $\{\vert z \vert < 1 \}$. We will use the spherical metric throughout, although you can replace it with the Euclidean metric as long as the objects you are dealing with is away from $\infty$. Denote by $\mathbb{D}(z,r)$ the open disk centered at $z$ of radius $r>0$ in the spherical metric.

Definition & Theorem: Let $Z$ be a Jordan curve in $\hat{\mathbb{C}}$, and let $X$ and $Y$ be the components of $\hat{\mathbb{C}} \backslash Z$. Then, Z is a quasicircle if it satisfies any of the following equivalent definitions:

1. There is a $K$-quasiconformal homeomorphism $\phi: \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ mapping the unit circle $\mathbb{T}$ onto $Z$;
2. There is some $K\geq 1$ such that any Riemann mapping $f: X \to \mathbb{D}$ extends to a $K$-quasiconformal map of $\hat{\mathbb{C}}$;
3. There is some $K\geq 1$ such that for every $x_1, x_2 \in X$, there is a $K$-quasiconformal map $\psi : \hat{\mathbb{C}} \to \hat{\mathbb{C}}$ such that $\psi(X)=X$ and $\psi(x_1)=x_2$;
4. There is some $K\geq 1$ such that for any four cyclically ordered distinct points $z_1, z_2, z_3, z_4 \in Z$, if the quadrilateral $R_x = X(z_1,z_2,z_3,z_4)$ has modulus $\text{mod}(R_x) = 1$ then the quadrilateral $R_y = Y(z_4,z_3,z_2,z_1)$ has modulus $\text{mod}(R_y) \leq K$;
5. There is some $c\geq 1$ such that if $I$ and $J$ are two disjoint closed intervals on $Z$ and if $\Gamma_{\hat{\mathbb{C}}}$ and $\Gamma_{X}$ are the families of all paths connecting $I$ and $J$ in $\hat{\mathbb{C}}$ and $X$ respectively, the extremal lengths $\lambda(\Gamma_{\hat{\mathbb{C}}})$ and $\lambda(\Gamma_X)$ satisfy $\lambda(\Gamma_{\hat{\mathbb{C}}}) \leq 2c \cdot \lambda(\Gamma_X)$.
6. If $f : X \to \mathbb{D}$ and $g : Y \to \hat{\mathbb{C}} \backslash \bar{\mathbb{D}}$ are conformal isomorphisms, considering their extensions to the boundary, the composition $h = g \circ f^{-1}$ is a quasisymmetric map on the unit circle $\mathbb{T}$: $h$ is a homeomorphism where there is some $c\geq 1$ such that for any $z_1,z_2,z_3 \in \mathbb{T}$, \(\frac{d(h(x_1),h(x_2))}{d(h(x_1),h(x_3))} \leq c \frac{d(x_1,x_2)}{d(x_1,x_3)}.\)
7. For any $x \in X$ and $y \in Y$, the harmonic measure $\omega_X(x, \cdot)$ of $Z$ in $X$ about $x$ is quasisymmetrically equivalent to the harmonic measure $\omega_Y(y, \cdot)$. More precisely, there is some $c\geq 1$ such that every two adjacent intervals $I, J \subset Z$ satisfy \(\frac{\omega_X(x,I)}{\omega_X(x,J)} \leq c \frac{\omega_Y(y,I)}{\omega_Y(y,J)};\)
8. $Z$ satisfies the bounded turning condition: there is some $c \geq 1$ such that for any two distinct points $x,y \in Z$, if $\gamma_1$ and $\gamma_2$ are the two components of $Z \backslash \{x,y\}$, then $\min_{i=1,2} \text{diam}(\gamma_i) \leq c \cdot d(x,y)$;
9. $Z$ is linearly locally connected (LLC): there is some $c \geq 1$ such that for every $z \in Z$ and $r>0$, any two points in $Z \cap \overline{\mathbb{D}(z,r)}$ can be joined by an interval in $Z \cap \overline{\mathbb{D}(z,cr)}$, and any two points in $Z \backslash \mathbb{D}(z,r)$ can be joined by an interval in $Z \backslash \mathbb{D}(z,r/c)$;
10. There is some $a>0$ such that any holomorphic function $f: X \to \mathbb{C}$ satisfying $\vert S_f \vert \leq a \cdot \rho_X^2$ on X, where $S_f = \left( \frac{f’’}{f’}\right)’ - \frac{1}{2}\left( \frac{f’’}{f’}\right)^2$ is the Schwarzian derivative of $f$ and $\rho_X(z) dz$ is the hyperbolic metric on $X$, is injective.

The constants $K$ and $c$ above are in general different for each property, but they do depend on each other. It is not difficult to show that the minimal constant $K=1$ and $c=1$ for properties 1-7 and 9 are achieved by circles. I would like to point out that there are many many other equivalent ways of defining quasicircles, or perhaps quasidisks (i.e. Jordan domains bounded by a quasicircle). Gehring published a paper² on a list of more than 20 different characterizations of quasicircles. Below, I will give non-detailed explanations on the intuition behind the 10 listed above.

Through quasiconformal maps

The typical definition of quasicircles is 1. Recall that quasiconformal maps locally distort angles up to some finite constant. This means that unlike a circle, $Z$ is allowed to have angles at each point but these angles must be definite. In particular, $Z$ should contain no cusps! Moreover, since quasiconformal maps map sets of zero measure to sets of zero measure, every quasicircle must have zero area. (Yes! Jordan curves of positive area do exist!)

From 1, property 2 is based on the fact that $g = f \circ \phi^{-1}: \mathbb{D}$ is a quasiconformal self map of the unit disk $\mathbb{D}$ and it has a quasiconformal extension to the whole $\hat{\mathbb{C}}$ that is $\mathbb{T}$-symmetric, so then $f$ extends to $\phi \circ g$ on $\hat{\mathbb{C}}$. From 2, property 3 satisfied by $\psi=f^{-1}\circ g\circ f$ where $g$ is a Möbius transformation which sends $f(x_1)$ to $f(x_2)$.

Refer to this post for the definition of topological quadrilaterals. Property 2 implies property 4: if $f: X \to \mathbb{D}$ is a Riemann mapping, then by the symmetry of $\partial \mathbb{D}$ and by conformal invariance of the modulus, $\text{mod}(f(R_y)) = \text{mod}(f(R_x)) = \text{mod}(R_x) = 1$, but then by property 2, $\text{mod}(R_y) \leq K \text{mod}(f(R_y))$.

Through quasisymmetries

Property 6 is often called conformal welding or sewing, and the quasisymmetric homeomorphism $h$ is called the sewing homeomorphism. There is a strong connection between quasisymmetries and quasiconformal maps. In general, quasisymmetries are homeomorphisms on a metric space which distort cross ratios in a controlled way. Any quasiconformal map on $\mathbb{D}$ extends to a quasisymmetric self map on $\mathbb{T}$. Conversely, any quasisymmetric self map on $\mathbb{T}$ admits a continuous extension to a quasiconformal self map on $\mathbb{D}$. The extension is not canonical, but some of the commonly used ones are due to Ahlfors-Beurling and Douady-Earle.

Through harmonic measures

It is not difficult to see the equivalence between 6 and 7. (You may want to look at this post if you want to know more about harmonic measures.) Property 7 indicates that given an interval $I$ on $Z$, it is not possible for the harmonic measure of $I$ on $X$ to be arbitrarily small when that of $I$ on $Y$ is definite, and vice versa. This happens, for example, $I$ represents a cusp pointing out of $X$ and towards $Y$. This is because the probability for brownian motion inside a topological disk to land near an outward cusp would be much smaller compared landing near an inward cusp.

Through basic geometric analysis

The bounded turning condition (property 8) is another popular characterisation because it is the easiest to state. Notice that this condition fails near a cusp or a ‘bottleneck’ where two points can be arbitrarily close to one another but the diameter of the shortest interval on $Z$ joining these two points decreases much more slowly. The equivalence between 1 and 8 is proven in the book of Ahlfors. With some elementary analysis, one can show that the bounded turning condition is equivalent to the LLC condition (property 9). (See Lehto’s book³ for more details.)

Through univalent functions

You may think that property 10 seems to be the most obscure one. Let me give you some intuition. You can check that the Schwarzian derivative of any Möbius transformation is always constantly zero, and any function with zero Schwarzian derivative is a Möbius transformation. The Schwarzian derivative $S_f$ of a univalent function $f$ measures how far $f$ is from a Möbius transformation. For any univalent function $f$ on $X$, $\vert S_f \vert \rho_X^{-2} \leq 3$ on $X$ and Krauss and Nehari proved that the constant $3$ is sharp. The constant $a$ in property 10 is often called the inner radius of univalence or the Schwarzian radius of injectivity. The proof that property 10 characterises quasicircles is much more involved than the other properties mentioned. I’ll personally recommend seeing Lehto’s book³ for more details.

References

1: L. Ahlfors. Lectures on Quasiconformal Mappings. Van Nostrand, 1966.
2: F. W. Gehring. Characterizations of Quasidisks. Banach Center Publications, Vol 48, 1999.
3: O. Lehto. Univalent functions and Teichmüller spaces, Springer-Verlag, 1987. ——