conformal radius


Definition.

Let G be a simply connected region that is not the whole plane and let aG be any point. The Riemann mapping theoremMathworldPlanetmath tells us that there exists a unique one-to-one and onto holomorphic map f:𝔻G (where 𝔻 is the unit disc) such that f(0)=a and f(0)>0. Then define the conformal radiusMathworldPlanetmath r(G,a)=f(0).

Example.

For example, take G=B(0,δ) (the open ball of radius δ around 0) for some δ>0, then r(G,0)=δ because we have a map f(z)=δz as our unique map. And thus this definition coincides with our definition of radius for this special case.

Example.

For another example we look at how the conformal radius is affected by the choice of the point a. So suppose that we take G to be the unit disc (D) itself and we take some point aD. The unique map that takes 0 to a is the map f(z)=z+a1+a¯z (where a¯ is the complex conjugateDlmfMathworldPlanetmath of a) and by the quotient rule we get that f(z)=1-|a|2(1+a¯z)2. And so r(D,a)=f(0)=1-|a|2, so the conformal radius of the unit disc goes to 0 as we move the point a towards the boundary of the disc, and it is largest (equal to 1) when a=0.

From the first example we can now see another way of characterizing the conformal radius. Take the inverse map (inverses of holomorphic one-to-one functions are also always holomorphic) and call it φ:G𝔻 (the map such that φ(f(z))=z). We take the derivative (see the entry on univalent functionsMathworldPlanetmath (http://planetmath.org/UnivalentFunction)) we get φ(f(0))=1f(0), that is φ(a)=1r (where we call r=r(G,a) for brevity now). If we multiply the map by the conformal radius we get a map γ:GB(0,r) such that γ(z)=rφ(z) and γ(a)=1. By uniqueness of the map arising from the Riemann mapping theorem we can see that γ is also unique. Thus we could define the conformal radius as follows.

Definition.

Let G be a region and let aG be any point. By application of Riemann mapping theorem there exists a unique map γ:GB(0,r) for some r>0, such that γ(a)=0 and γ(a)=1. The conformal radius is then defined as r(G,a)=r.

This definition gives more of an intuitive understanding of why we’d call this the conformal radius of G. We look at the unique map with γ(a)=1, that is, the map that doesn’t “stretch” the set. So the radius of G with respect to a is really the radius of the unique ball around zero to which G is conformally equivalent without any “stretching” needed.

References

  • 1 S. Rohde, M. Zinsmeister. , Journal d’Analyse (to appear). Available at http://www.math.washington.edu/ rohde/papers/rozi.pshttp://www.math.washington.edu/ rohde/papers/rozi.ps
Title conformal radius
Canonical name ConformalRadius
Date of creation 2013-03-22 14:18:33
Last modified on 2013-03-22 14:18:33
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 9
Author jirka (4157)
Entry type Definition
Classification msc 30C55
Related topic RiemannMappingTheorem