Math 213a: Complex analysis Problem Set #8 (12 November 2003): Harmonic functions and their uses, cont’d

First, an observation on the coefficients of the linear equations used to determine the logarithm of our conformal map of a finitely connected region (Ahlfors, V.3.1, Ex.2 (p.204)):

1. Let Ω ⊂ C be a region whose boundary consists of the simple closed curves C_{1}, …, C_{n}. As in Ahlfors V.3.1, for j, k ∈ {1, …, n} let α_{jk} = ∮_{C_{j}} ∗dω_{k}, where ω_{k} is the harmonic measure of C_{k} with respect to Ω, and C_{j} is traversed in the positive direction relative to Ω. Prove that α_{jk} = α_{kj}.

Next, some problems on the nice special case of a doubly-connected region. We begin with an elementary problem (that is, a problem not requiring the machinery of (sub)harmonic functions from Chapter V) that picks up a thread from the fourth problem set, where we studied the action of PGL_{2}(C) on pairs of disjoint circles.

2. If Ω is the doubly connected region bounded by two disjoint circles, show that the conformal bijections of Ω to an annulus are fractional linear transformations. Deduce that the subgroup of PGL_{2}(C) that preserves two disjoint circles is conjugate in PGL_{2}(C) to the group of fractional linear transformations of the form z ↦ cz for some c ∈ C^{∗} with |c| = 1. Obtain a similar description of the stabilizer in PGL_{2}(C) of two circles that are tangent or meet in two points.

3. Let Ω be the annulus {z ∈ C : r < |z| < R}. Prove that if u is a bounded harmonic function on Ω then, for any ρ ∈ (r, R),

(log \frac{R}{r}) ∮_{|z|=ρ} ∗du ≤ 2π(sup_{z} u(z) − inf_{z} u(z)),

with equality if and only if there exist C_{0}, C_{1} ∈ R, with C_{1} ≥ 0, such that u(z) = C_{0} + C_{1} log |z| for all z ∈ Ω. Conclude that the only conformal bijections of Ω are z ↦ cz and z ↦ cRr/z (|c| = 1).

4. Suppose Ω and Ω′ ∈ C are doubly connected regions conformally equivalent to the annuli r < |z| < R and r′ < |z| < R′. Let their complements in P^{1}(C) have components E_{1}, E_{2} and E′_{1}, E′_{2}, with E_{1}, E′_{1} containing ∞. Prove that if E_{j} ⊇ E′_{j} for j = 1, 2 (whence Ω ⊆ Ω′) then R/r ≤ R′/r′, with equality only when Ω = Ω′.

5. Suppose Ω is a simply-connected bounded subset of C, and fix z_{0} ∈ Ω. For each r > 0 such that Ω contains the closed disc {z : |z−z_{0}| ≤ r}, the region Ω_{r} := {z ∈ Ω : |z − z_{0}| > r} is doubly connected, and thus conformal with an annulus A_{r} with outer radius 1. Let ρ(r) be the inner radius of A_{r}. By the previous problem, ρ is a decreasing function of r. Prove that ρ(r) → 0 as r → 0. What happens to the conformal bijections from Ω_{r} to A_{r} as r → 0?

Finally:

6. Let Ω ∈ C be a region of connectivity n.

i) If n ≤ 3, prove that Ω is conformally equivalent to its image under complex conjugation. Do this even in the (easier) case that the complement of Ω in P^{1}(C) has one or more components that reduce to a point.

ii^{∗}) When n = 4, construct or prove the existence of an Ω not conformally equivalent to its complex conjugate, with none of the components of P^{1}(C) − Ω reducing to a point.

This problem set is due Wednesday, November 19, at the beginning of class.