Logarithmic Capacity

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'', 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

: $\mathrm(z,D) := \frac\,.$

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 $\mathrm(\varphi(z),D') = , \varphi'(z), \, \mathrm(z,D)$.

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

: $\mathrm(z,D) = \frac\,.$

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

:$f(z)=i\frac,$

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,

:$\frac \leq \mathrm (z,\partial D) \leq \mathrm(z,D),$

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

Version from infinity: transfinite diameter and logarithmic capacity

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

: $\mathrm(\infty,D) := \frac := \lim_ \frac,$

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

:$f(z)=c_1z+c_0 + c_z^ + \dots, \qquad c_1\in\mathbf_+.$

The coefficient ''c''₁ = rad(∞, ''D'') equals the transfinite diameter and the (logarithmic) capacity of ''D''; see Chapter 11 of and . See also the article on the capacity of a set.

The coefficient ''c''₀ is called the conformal center of ''D''. It can be shown to lie in the convex hull of ''D''; moreover,

:$D\subseteq \\,,$

where the radius 2''c''₁ is sharp for the straight line segment of length 4''c''₁. See pages 12–13 and Chapter 11 of .

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

:$d(z_1,\ldots,z_k):=\prod_ , z_i-z_j,$

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

:$d_n(D):=\sup_ d(z_1,\ldots,z_n)^$

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 $d(D):=\lim_ d_n(D)$ exists and it is called the Fekete constant.

Now let $\mathcal P_n$ denote the set of all monic polynomials of degree ''n'' in C 'x'' let $\mathcal Q_n$ denote the set of polynomials in $\mathcal P_n$ with all zeros in ''D'' and let us define

:$\mu_n(D):=\inf_ \sup_ , p(z),$ and $\tilde_n(D):=\inf_ \sup_ , p(z),$

Then the limits

:$\mu(D):=\lim_ \mu_n(D)^\frac1n$ and $\mu(D):=\lim_ \tilde_n(D)^\frac1n$

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 .

References

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Further reading

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External links

*. From MathWorld — A Wolfram Web Resource, created by Eric W. Weisstein.

{{DEFAULTSORT:Conformal Radius
Complex analysis