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

: $\backslash operatorname\{rad\}(z,D)\; :=\; \backslash frac\{1\}\{f\text{'}(z)\}\backslash ,.$

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 $\backslash operatorname\{rad\}(\backslash varphi(z),D\text{'})\; =\; |\backslash varphi\text{'}(z)|\backslash operatorname\{rad\}(z,D)$.

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

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

: $\backslash operatorname\{rad\}(z,D)\; =\; \backslash frac\{2\backslash operatorname\{Im\}(g(z))\}$

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

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

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,

:$\backslash frac\{\backslash operatorname\{rad\}(z,D)\}\{4\}\; \backslash leq\; \backslash operatorname\{dist\}\; (z,\backslash partial\; D)\; \backslash leq\; \backslash operatorname\{rad\}(z,D),$

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

{{main|Analytic capacity}}

{{main|Capacity of a set}}

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 ∞{{citation needed|date=November 2020}}, and one can define

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

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

:$f(z)=c\_1z+c\_0\; +\; c\_\{-1\}z^\{-1\}\; +\; \backslash cdots,\; \backslash qquad\; c\_1\backslash in\backslash mathbf\{R\}\_+.$

The coefficient c₁ = rad(∞, D) equals the transfinite diameter and the (logarithmic) capacity of D; see Chapter 11 of {{harvtxt|Pommerenke|1975}} and {{harvtxt|Kuz′mina|2002}}.

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

:$D\backslash subseteq\; \backslash \{z:\; |z-c\_0|\backslash leq\; 2\; c\_1\backslash \}\backslash ,,$

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

:

denote the product of pairwise distances of the points $z\_1,\backslash ldots,z\_k$ and let us define the following quantity for a compact set D ⊂ C:

:$d\_n(D):=\backslash sup\_\{z\_1,\backslash ldots,z\_n\backslash in\; D\}\; d(z\_1,\backslash ldots,z\_n)^\{1\backslash left/\backslash binom\; n\; 2\backslash right.\}$

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 $d(D):=\backslash lim\_\{n\backslash to\backslash infty\}\; d\_n(D)$ exists and it is called the Fekete constant.

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

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

Then the limits

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

exist and they are called the Chebyshev constant and modified Chebyshev constant, respectively.

Michael Fekete and Gábor Szegő proved that these constants are equal.

The conformal radius is a very useful tool, e.g., when working with the Schramm–Loewner evolution. A beautiful instance can be found in {{harvtxt|Lawler|Schramm|Werner|2002}}.

{{sfn whitelist|CITEREFKuz′mina2002}}

• {{cite book| last=Ahlfors | first=Lars V. | author-link=Lars Ahlfors | year=1973 | title=Conformal invariants: topics in geometric function theory | url=https://archive.org/details/conformalinvaria00lars | url-access=registration | series=Series in Higher Mathematics | publisher=McGraw-Hill | isbn=978-0-07-000659-1 | mr=0357743 | zbl=0272.30012 }}
• {{cite book| editor1-last=Horváth | editor1-first=János | year=2005 | title=A Panorama of Hungarian Mathematics in the Twentieth Century, I | series=Bolyai Society Mathematical Studies | publisher=Springer | isbn=3-540-28945-3 }}
• {{SpringerEOM | last1=Kuz′mina | first1=G. V.|year=2002|title=Conformal radius of a domain}}
• {{Citation | last1=Lawler | first1=Gregory F. |author-link1=Gregory Lawler| last2=Schramm | first2=Oded | author-link2=Oded Schramm| last3=Werner | first3=Wendelin | author-link3=Wendelin Werner|title= One-arm exponent for critical 2D percolation | url= http://www.math.washington.edu/~ejpecp/viewarticle.php?id=1303&layout=abstract | mr=1887622 | year=2002 | journal= Electronic Journal of Probability | volume=7 | issue=2 | pages=13 pp | zbl=1015.60091| issn=1083-6489 | doi=10.1214/ejp.v7-101| arxiv=math/0108211 }}
• {{cite book| last=Pommerenke | first= Christian | author-link=Christian Pommerenke | year=1975 | title=Univalent functions | others=With a chapter on quadratic differentials by Gerd Jensen | publisher=Vandenhoeck & Ruprecht | location=Göttingen | zbl=0298.30014 | series=Studia Mathematica/Mathematische Lehrbücher | volume=Band XXV }}

• {{citation | last=Rumely | first=Robert S. | author-link = Robert Rumely | title=Capacity theory on algebraic curves | series=Lecture Notes in Mathematics | volume=1378 | location=Berlin etc. | publisher=Springer-Verlag | year=1989 | isbn=3-540-51410-4 | zbl=0679.14012 }}

• {{Citation | last1=Pooh | first1=Charles | title=Conformal radius| url= http://mathworld.wolfram.com/ConformalRadius.html}}. From MathWorld — A Wolfram Web Resource, created by Eric W. Weisstein.

{{DEFAULTSORT:Conformal Radius}}

Category:Complex analysis

Category:Radii