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.
Given a simply connected domain D ⊂ C, and a point z ∈ D, by the Riemann mapping theorem there exists a unique conformal map f : D → D onto the unit disk (usually referred to as the uniformizing map) with f(z) = 0 ∈ D and f′(z) ∈ R+. The conformal radius of D from z is then defined as
\operatorname{rad}(z,D):=
| 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 ↦ x/r. See below for more examples.
One reason for the usefulness of this notion is that it behaves well under conformal maps: if φ : D → D′ is a conformal bijection and z in D, then
\operatorname{rad}(\varphi(z),D')=|\varphi'(z)|\operatorname{rad}(z,D)
The conformal radius can also be expressed as
\exp(\xix(x))
\xix(y)
log(|x-y|)
\partialD
D
Let K ⊂ H be a subset of the upper half-plane such that D := H\K is connected and simply connected, and let z ∈ 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 → H. Then, for any such map g, a simple computation gives that
\operatorname{rad}(z,D)=
| 2\operatorname{Im | |
| (g(z))}{|g'(z)|}. |
For example, when K = ∅ and z = i, then g can be the identity map, and we get rad(i, H) = 2. Checking that this agrees with the original definition: the uniformizing map f : H → D is
| f(z)=i | z-i |
| z+i |
,
and then the derivative can be easily calculated.
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 ∈ D ⊂ C,
| \operatorname{rad | |
| (z,D)}{4} |
\leq\operatorname{dist}(z,\partialD)\leq\operatorname{rad}(z,D),
where dist(z, ∂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 taking D = D and z = 0.
The lower bound is attained by the following “slit domain”: D = C\R+ and z = −r ∈ R−. The square root map φ takes D onto the upper half-plane H, with
\varphi(-r)=i\sqrt{r}
| |\varphi'(-r)|= | 1 |
| 2\sqrt{r |
\operatorname{rad}(i\sqrt{r},H)=2\sqrt{r}
See main article: Analytic capacity.
See main article: Capacity of a set. When D ⊂ C is a connected, simply connected compact set, then its complement E = Dc is a connected, simply connected domain in the Riemann sphere that contains ∞, and one can define
\operatorname{rad}(infty,D):=
| 1 | |
| \operatorname{rad |
(infty,E)}:=\limz\toinfty
| f(z) | |
| z |
,
where f : C\D → E is the unique bijective conformal map with f(∞) = ∞ and that limit being positive real, i.e., the conformal map of the form
f(z)=c1z+c0+c-1z-1+ … , c1\inR+.
The coefficient c1 = rad(∞, D) equals the transfinite diameter and the (logarithmic) capacity of D; see Chapter 11 of and .
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-c0|\leq2c1\},
where the radius 2c1 is sharp for the straight line segment of length 4c1. See pages 12–13 and Chapter 11 of .
We define three other quantities that are equal to the transfinite diameter even though they are defined from a very different point of view. Let
d(z1,\ldots,zk):=\prod1\le|zi-zj|
denote the product of pairwise distances of the points
z1,\ldots,zk
dn(D):=\sup
| z1,\ldots,zn\inD |
d(z1,\ldots,z
| 1\left/\binomn2\right. | |
| n) |
In other words,
dn(D)
The limit
d(D):=\limn\toinftydn(D)
Now let
lPn
lQn
lPn
\mun(D):=inf
| p\inlPn |
\supz\in|p(z)|
\tilde{\mu}n(D):=inf
| p\inlQn |
\supz\in|p(z)|
Then the limits
\mu(D):=\limn\toinfty
| 1/n | |
| \mu | |
| n(D) |
\mu(D):=\limn\toinfty
| 1/n | |
| \tilde{\mu} | |
| n(D) |
exist and they are called the Chebyshev constant and modified Chebyshev constant, respectively.Michael Fekete and Gábor Szegő proved that these constants are equal.
The conformal radius is a very useful tool, e.g., when working with the Schramm–Loewner evolution. A beautiful instance can be found in .