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 ⊂ C, and a point z ∈ D, it then follows from the Riemann mapping theorem that there exists a unique conformal homeomorphism f : D → D onto the open 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

$\mathrm{rad}(z,D):=\frac{1}{f^{\prime }(z)}.$

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 $\mathrm{rad}(\phi (z),D^{\prime })=|{\phi }^{\prime }(z)|\mathrm{rad}(z,D)$.

The conformal radius can also be expressed as $\exp ({\xi }_x(x))$ where ${\xi }_x(y)$ is the harmonic extension of $\log (|x-y|)$ from $\partial D$ to $D$.

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

$\mathrm{rad}(z,D)=\frac{2\mathrm{Im}(g(z))}{|g^{\prime }(z)|}.$

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

$f(z)=i\frac{z-i}{z+i},$

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 ∈ D ⊂ C,

$\frac{\mathrm{rad}(z,D)}{4}\le \mathrm{dist}(z,\partial D)\le \mathrm{rad}(z,D),$

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 $\phi (-r)=i\sqrt{r}$ and derivative $|{\phi }^{\prime }(-r)|=\frac{1}{2\sqrt{r}}$. The above formula for the upper half-plane gives $\mathrm{rad}(i\sqrt{r},ℍ)=2\sqrt{r}$, 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

Template:Main Template:Main When D ⊂ C is a connected, simply connected compact set, then its complement E = D^c is a connected, simply connected domain in the Riemann sphere that contains ∞,Template:Citation needed and one can define

$\mathrm{rad}(\mathrm{\infty },D):=\frac{1}{\mathrm{rad}(\mathrm{\infty },E)}:=\underset{z\to \mathrm{\infty }}{lim}\frac{f(z)}{z},$

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

$f(z)=c_{1}z+c_0+c_{-1}{z}^{-1}+\cdots ,c_1\in {𝐑}_{+}.$

The coefficient c₁ = rad(∞, D) equals the transfinite diameter and the (logarithmic) capacity of D; see Chapter 11 of Template:Harvtxt and Template:Harvtxt.

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

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

where the radius 2c₁ is sharp for the straight line segment of length 4c₁. See pages 12–13 and Chapter 11 of Template: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

$d(z_1,\dots ,z_k):=\prod _{1\le i<j\le k}|z_i-z_j|$

denote the product of pairwise distances of the points $z_1,\dots ,z_k$ and let us define the following quantity for a compact set D ⊂ C:

$d_n(D):=\underset{z_1,\dots ,z_n\in D}{sup}d(z_1,\dots ,z_n{)}^{1/(\genfrac{}{}{0ex}{}{n}{2})}$

In other words, $d_n(D)$ 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 $d(D):=\underset{n\to \mathrm{\infty }}{lim}{d}_n(D)$ exists and it is called the Fekete constant.

Now let ${𝒫}_n$ denote the set of all monic polynomials of degree n in C[x], let ${𝒬}_n$ denote the set of polynomials in ${𝒫}_n$ with all zeros in D and let us define

${\mu }_n(D):=\underset{p\in {𝒫}_n}{inf}\underset{z\in D}{sup}|p(z)|$ and ${\tilde{\mu }}_n(D):=\underset{p\in {𝒬}_n}{inf}\underset{z\in D}{sup}|p(z)|$

Then the limits

$\mu (D):=\underset{n\to \mathrm{\infty }}{lim}{\mu }_n(D{)}^{1/n}$ and $\mu (D):=\underset{n\to \mathrm{\infty }}{lim}{\tilde{\mu }}_n(D{)}^{1/n}$

exist and they are called the Chebyshev constant and modified Chebyshev constant, respectively. Michael Fekete and Gábor Szegő proved that these constants are equal.Template:Citation needed

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 Template:Harvtxt.

References

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Further reading

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External links

• Template:Citation. From MathWorld — A Wolfram Web Resource, created by Eric W. Weisstein.