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In another question here on MO, Anweshi asks if any doubly connected region in the complex plane can be conformally mapped to some annulus. The answer to this is yes. But the fact is that two annuli are conformally equivalent iff the ratio of the outer radius and the inner radius is the same for the two. Thus each is conformally equivalent to a unique "standard" annulus $r < |z| < 1$ with $ 0 < r < 1$. Now my question is the following:

Is there any way to see what the radius of a "standard" annulus conformally equivalent to a doubly connected region is, just by looking at the region? That is without constructing an explicit map.

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This really depends on what you mean by "looking at" a region, but there are integrals that show up in conformal field theory that allow you to distinguish conformally inequivalent surfaces. – S. Carnahan Jan 9 '10 at 18:25
re: Scott Carnahan's comment. see e.g. this section in Lawler's book:… – j.c. Jan 9 '10 at 18:28
Or see the wikipedia article on extremal length at . – Harald Hanche-Olsen Jan 9 '10 at 18:57
Ah, nice. Thank you very much! – Grétar Amazeen Jan 9 '10 at 19:06
up vote 14 down vote accepted

By "see" I will assume you mean in a geometric sense. Then your question falls within a standard topic in geometric complex analysis. First some terminology: A doubly connected domain $R$ on the Riemann sphere is called a ring domain, and if you map it onto $r < |z| < s$ as a canonical domain, then $\mathrm{mod}(R) = (2\pi)^{-1}\log(s/r)$ is called the conformal modulus or just modulus of the ring domain. By the way I have defined it, it is nearly trivially a conformal invariant, but it need not be defined this way. There is a geometric theory, the Ahlfors-Beurling theory of extremal length of curve families, within which the modulus of a ring domain can be defined directly and geometrically, without any preliminary conformal mapping onto some canonical domain. Extremal length can be proved to be a conformal invariant, and then one quickly sees that the two definitions coincide. There is an exposition of the theory of extremal length in Conformal Invariants by Ahlfors.

It would be unreasonable to expect to "see" the exact value of the modulus of a ring domain. The boundary of a ring domain can be extremely complicated geometrically, and every tiny wiggle impacts on the modulus. But extremal length yields inequalities for the modulus from geometric data. I will quote a single, rather striking, result of this kind:

If a ring domain $R$ contains no circle on the Riemann sphere separating its two boundary components, then $\mathrm{mod}(R) \leq 1/4$. The constant $1/4$ is sharp. The result is due to D. A. Herron, X. Y. Liu and D. Minda.

Small modulus means "thin" ring domain; if the modulus is large enough, the ring domain is so "fat" that it has to contain a separating circle.

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