Conformal radius

In mathematics, the conformal radius is a way to measure the size of a simply connected planar domain D viewed from a point z in it. As opposed to notions using Euclidean distance (say, the radius of the largest inscribed disk with center z), this notion is well-suited to use in complex analysis, in particular in conformal maps and conformal geometry.

A closely related notion is the transfinite diameter or (logarithmic) capacity of a compact simply connected set D, which can be considered as the inverse of the conformal radius of the complement E = D^c viewed from infinity.

Definition

Given a simply connected domain $D \subset \C$ in the complex plane, and a point $z\in D$, by the Riemann mapping theorem there exists a unique conformal map $f: D\to\mathbb D$ onto the unit disk with $f(z)=0\in \mathbb D$ and derivative $f'(z)\in \R_+$. (This is usually called the uniformizing map.) The conformal radius of D from z is then defined as

$\mathsf{rad}(z,D) := \frac{1}{f'(z)}\,.$

The simplest example is that the conformal radius of the disk of radius r viewed from its center is also r, shown by the uniformizing map $x\mapsto x/r$. See below for more examples.

One reason for the usefulness of this notion is that it behaves well under conformal maps: if $\varphi: D \to D'$ is a conformal bijection and $z\in D$, then $\mathsf{rad}(z',D') = |\varphi'(z)|\, \mathsf{rad}(z,D)$.

A special case: the upper-half plane

Let $K\subset \mathbb H$ be a subset of the upper half-plane such that $D:=\mathbb{H} \setminus K$ is connected and simply connected, and let $z\in D$ be a point. (This is a usual scenario, say, in the Schramm-Loewner evolution). By the Riemann mapping theorem, there is a conformal bijection $g: D \to\mathbb H$. Then, for any such map g, a simple computation gives that

$\mathsf{rad}(z,D) = \frac{2\, \mathsf{Im}(g(z))}{|g'(z)|}\,.$

For example, when $K=\emptyset$ and z = i, then g can be the identity map, and we get $\mathsf{rad}(i,\mathbb{H})=2$. Checking that this agrees with the original definition: the uniformizing map $f: \mathbb{H} \to\mathbb D$ is $f(z)=i\frac{z-i}{z+i}$, and then the derivative can be easily calculated.

Relation to inradius

That it is a good measure of radius is shown by the following immediate consequence of the Schwarz lemma and the Koebe 1/4 theorem: for $z\in D\subset \C$,

$\frac{\mathsf{rad}(z,D)}{4} \leq \mathsf{dist} (z,\partial D) \leq \mathsf{rad}(z,D)\,,$

where $\mathsf{dist} (z,\partial D)$ denotes the Euclidean distance between z and the boundary of D, or in other words, the radius of the largest inscribed disk with center z.

Both inequalities are best possible:

The upper bound is clearly attained by the unit disk with $z=0\in\mathbb D$ the origin.

The lower bound is attained by the following “slit domain”: $D=\mathbb{C}\setminus \mathbb{R}_+$ and $z=-r\in \mathbb{R}_-$. The square root map φ takes D onto the upper half-plane $\mathbb H$, with $\varphi(-r) = i\sqrt{r}$ and derivative $|\varphi'(-r)|=\frac{1}{2\sqrt{r}}$. The above formula for the upper half-plane gives $\mathsf{rad}(i\sqrt{r},\mathbb{H})=2\sqrt{r}$, and then the formula for transformation under conformal maps gives $\mathsf{rad}(-r,D)=4r$, while, of course, $\mathsf{dist}(-r,\partial D)=r$.

Version from infinity: transfinite diameter and logarithmic capacity

When $D\subset \C$ is a simply connected compact set, then its complement E = D^c is a simply connected domain in the Riemann sphere that contains $\infty$, and one can define

$\mathsf{rad}(\infty,D) := \frac{1}{\mathsf{rad}(\infty,E)} := \lim_{z\to\infty} f(z)/z\,,$

where$f: \C \setminus \mathbb D \to E$ is the unique bijective conformal map with $f(\infty)=\infty$ and that limit being positive real, i.e., the conformal map of the form

$f(z)=c_1z+c_0 + c_{-1}z^{-1} + \dots\,,\$ with $c_1\in\R_+\,.$

The coefficient $c_1=\mathsf{rad}(\infty,D)$ equals the transfinite diameter and the (logarithmic) capacity of D; see Chapter 11 of Pommerenke (1975) and Kuz′mina (2002). See also the article on the capacity of a set.

The coefficient c₀ is called the conformal center of D. It can be shown to lie in the convex hull of D; moreover,

$D\subseteq \{z: |z-c_0|\leq 2 c_1\}\,,$

where the radius 2c₁ is sharp for the straight line segment of length 4c₁. See pages 12–13 and Chapter 11 of Pommerenke (1975).

Applications

The conformal radius is a very useful tool, e.g., when working with the Schramm-Loewner evolution. A beautiful instance can be found in Lawler, Schramm & Werner (2002).

References

• Ahlfors, Lars V. (1973). Conformal invariants: topics in geometric function theory. Series in Higher Mathematics. McGraw-Hill. MR0357743. 

• Kuz′mina, G. V. (2002), Conformal radius of a domain, http://eom.springer.de/c/c024800.htm , from the Encyclopaedia of Mathematics online.

• Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin (2002), "One-arm exponent for critical 2D percolation", Electron. J. Probab. 7 (2): 13 pp., MR1887622, http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1303&layout=abstract 

• Pommerenke, Christian (1975). Univalent functions. Vandenhoeck & Ruprecht, Göttingen. 

• Pooh, Charles, Conformal radius, http://mathworld.wolfram.com/ConformalRadius.html . From MathWorld — A Wolfram Web Resource, created by Eric W. Weisstein.