Английская Википедия: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.
Definition
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
- <math>\operatorname{rad}(z,D) := \frac{1}{f'(z)}\,.</math>
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 <math>\operatorname{rad}(\varphi(z),D') = |\varphi'(z)|\operatorname{rad}(z,D)</math>.
The conformal radius can also be expressed as <math>\exp(\xi_x(x))</math> where <math>\xi_x(y)</math> is the harmonic extension of <math>\log(|x-y|)</math> from <math>\partial D</math> to <math>D</math>.
A special case: the upper-half plane
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
- <math>\operatorname{rad}(z,D) = \frac{2\operatorname{Im}(g(z))}{|g'(z)|}\,.</math>
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
- <math>f(z)=i\frac{z-i}{z+i},</math>
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 ∈ D ⊂ C,
- <math>\frac{\operatorname{rad}(z,D)}{4} \leq \operatorname{dist} (z,\partial D) \leq \operatorname{rad}(z,D), </math>
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 <math>\varphi(-r) = i\sqrt{r}</math> and derivative <math>|\varphi'(-r)|=\frac{1}{2\sqrt{r}}</math>. The above formula for the upper half-plane gives <math>\operatorname{rad}(i\sqrt{r},\mathbb{H})=2\sqrt{r}</math>, and then the formula for transformation under conformal maps gives rad(−r, D) = 4r, while, of course, dist(−r, ∂D) = r.
Version from infinity: transfinite diameter and logarithmic capacity
Шаблон:Main Шаблон:Main 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 ∞Шаблон:Citation needed, and one can define
- <math>\operatorname{rad}(\infty,D) := \frac{1}{\operatorname{rad}(\infty,E)} := \lim_{z\to\infty} \frac{f(z)}{z},</math>
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
- <math>f(z)=c_1z+c_0 + c_{-1}z^{-1} + \cdots, \qquad c_1\in\mathbf{R}_+.</math>
The coefficient c1 = rad(∞, D) equals the transfinite diameter and the (logarithmic) capacity of D; see Chapter 11 of Шаблон:Harvtxt and Шаблон:Harvtxt.
The coefficient c0 is called the conformal center of D. It can be shown to lie in the convex hull of D; moreover,
- <math>D\subseteq \{z: |z-c_0|\leq 2 c_1\}\,,</math>
where the radius 2c1 is sharp for the straight line segment of length 4c1. See pages 12–13 and Chapter 11 of Шаблон:Harvtxt.
The Fekete, Chebyshev and modified Chebyshev constants
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
- <math>d(z_1,\ldots,z_k):=\prod_{1\le i<j\le k} |z_i-z_j|</math>
denote the product of pairwise distances of the points <math>z_1,\ldots,z_k</math> and let us define the following quantity for a compact set D ⊂ C:
- <math>d_n(D):=\sup_{z_1,\ldots,z_n\in D} d(z_1,\ldots,z_n)^{1\left/\binom n 2\right.}</math>
In other words, <math>d_n(D)</math> is the supremum of the geometric mean of pairwise distances of n points in D. Since D is compact, this supremum is actually attained by a set of points. Any such n-point set is called a Fekete set.
The limit <math>d(D):=\lim_{n\to\infty} d_n(D)</math> exists and it is called the Fekete constant.
Now let <math>\mathcal P_n</math> denote the set of all monic polynomials of degree n in C[x], let <math>\mathcal Q_n</math> denote the set of polynomials in <math>\mathcal P_n</math> with all zeros in D and let us define
- <math>\mu_n(D):=\inf_{p\in\mathcal P_n} \sup_{z\in D} |p(z)|</math> and <math>\tilde{\mu}_n(D):=\inf_{p\in\mathcal Q_n} \sup_{z\in D} |p(z)|</math>
Then the limits
- <math>\mu(D):=\lim_{n\to\infty} \mu_n(D)^{1/n}</math> and <math>\mu(D):=\lim_{n\to\infty} \tilde{\mu}_n(D)^{1/n}</math>
exist and they are called the Chebyshev constant and modified Chebyshev constant, respectively. Michael Fekete and Gábor Szegő proved that these constants are equal.
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 Шаблон:Harvtxt.
References
Further reading
External links
- Шаблон:Citation. From MathWorld — A Wolfram Web Resource, created by Eric W. Weisstein.
