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.
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
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 .
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
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
and then the derivative can be easily calculated.
Relation to inradius
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 and derivative . The above formula for the upper half-plane gives , 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
When D ⊂ C is a simply connected compact set, then its complement E = Dc is a simply connected domain in the Riemann sphere that contains ∞ (needs citation), and one can define
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
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,
where the radius 2c1 is sharp for the straight line segment of length 4c1. See pages 12–13 and Chapter 11 of Pommerenke (1975).
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
denote the product of pairwise distances of the points and let us define the following quantity for a compact set D ⊂ C:
In other words, 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 exists and it is called the Fekete constant.
Now let denote the set of all monic polynomials of degree n in C[x], let denote the set of polynomials in with all zeros in D and let us define
Then the limits
- Ahlfors, Lars V. (1973). Conformal invariants: topics in geometric function theory. Series in Higher Mathematics. McGraw-Hill. MR 0357743. Zbl 0272.30012.
- Horváth, János, ed. (2005). A Panorama of Hungarian Mathematics in the Twentieth Century, I. Bolyai Society Mathematical Studies. Springer. ISBN 3-540-28945-3.
- Kuz′mina, G. V. (2002), Conformal radius of a domain, from the Encyclopaedia of Mathematics online.
- Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin (2002), "One-arm exponent for critical 2D percolation", Electronic Journal of Probability, 7 (2): 13 pp., doi:10.1214/ejp.v7-101, ISSN 1083-6489, MR 1887622, Zbl 1015.60091
- Pommerenke, Christian (1975). Univalent functions. Studia Mathematica/Mathematische Lehrbücher. Band XXV. With a chapter on quadratic differentials by Gerd Jensen. Göttingen: Vandenhoeck & Ruprecht. Zbl 0298.30014.
- Rumely, Robert S. (1989), Capacity theory on algebraic curves, Lecture Notes in Mathematics, 1378, Berlin etc.: Springer-Verlag, ISBN 3-540-51410-4, Zbl 0679.14012