Conformal radius

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 = Dc viewed from infinity.

Contents

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 = Dc 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\,,

wheref: \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. 

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