Conformal radius

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

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

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 ↦ 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 ${\displaystyle \mathrm {rad} (\varphi (z),D')=|\varphi '(z)|\,\mathrm {rad} (z,D)}$.

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

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

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

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

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

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,

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

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 ${\displaystyle \varphi (-r)=i{\sqrt {r}}}$ and derivative ${\displaystyle |\varphi '(-r)|={\frac {1}{2{\sqrt {r}}}}}$. The above formula for the upper half-plane gives ${\displaystyle \mathrm {rad} (i{\sqrt {r}},\mathbb {H} )=2{\sqrt {r}}}$, and then the formula for transformation under conformal maps gives rad(−r, D) = 4r, while, of course, dist(−r, ∂D) = r.

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 ∞, and one can define

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

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

${\displaystyle f(z)=c_{1}z+c_{0}+c_{-1}z^{-1}+\dots ,\qquad c_{1}\in \mathbf {R} _{+}.}$

The coefficient c₁ = rad(∞, 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,

${\displaystyle D\subseteq \{z:|z-c_{0}|\leq 2c_{1}\}\,,}$

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

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

${\displaystyle d(z_{1},\ldots ,z_{k}):=\prod _{1\leq i<j\leq k}|z_{i}-z_{j}|}$

denote the product of pairwise distances of the points ${\displaystyle z_{1},\ldots ,z_{k}}$ and let us define the following quantity for a compact set D ⊂ C:

${\displaystyle d_{n}(D):=\sup _{z_{1},\ldots ,z_{n}\in D}d(z_{1},\ldots ,z_{n})^{\frac {1}{\binom {n}{2}}}}$

In other words, ${\displaystyle d_{n}(D)}$ 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 ${\displaystyle d(D):=\lim _{n\to \infty }d_{n}(D)}$ exists and it is called the Fekete constant.

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

${\displaystyle \mu _{n}(D):=\inf _{p\in {\mathcal {P}}_{n}}\sup _{z\in D}|p(z)|}$ and ${\displaystyle {\tilde {\mu }}_{n}(D):=\inf _{p\in {\mathcal {Q}}_{n}}\sup _{z\in D}|p(z)|}$

Then the limits

${\displaystyle \mu (D):=\lim _{n\to \infty }\mu _{n}(D)^{\frac {1}{n}}}$ and ${\displaystyle \mu (D):=\lim _{n\to \infty }{\tilde {\mu }}_{n}(D)^{\frac {1}{n}}}$

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

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).

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• Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin (2002), "One-arm exponent for critical 2D percolation", Electronic Journal of Probability, 7 (2): 13 pp., arXiv:math/0108211, doi:10.1214/ejp.v7-101, ISSN 1083-6489, MR 1887622, Zbl 1015.60091
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