[14] The span of multiply connected domains

Abstract

In this seminal paper, Schiffer introduced an important conformal invariant for multiply connected plane domains, the span. The definition is in terms of conformal mappings onto canonical slit domains, and there is a complementary characterization (almost literally), in terms of an extremal problem for area under a conformal mapping. While the foundational results on conformal mapping of multiply connected domains concern the existence and uniqueness of mappings onto various canonical domains, Schiffer’s paper is an early example of using the mappings as a tool in geometric function theory. Moreover, his study of the extremal problem is clearly influenced by his development of variational methods.

[14] Menahem Max Schiffer, “The span of multiply connected domains,” in Duke Mathematical Journal, Volume 10 (1943), 209–216.

© 1943 Duke University Press. All rights reserved. Reprinted by permission of the publisher. www.dukeupress.edu

Commentary on

• [14] The span of multiply connected domains, Duke Math. J. 10 (1943), 209–216.

In this seminal paper, Schiffer introduced an important conformal invariant for multiply connected plane domains, the span. The definition is in terms of conformal mappings onto canonical slit domains, and there is a complementary characterization (almost literally), in terms of an extremal problem for area under a conformal mapping. While the foundational results on conformal mapping of multiply connected domains concern the existence and uniqueness of mappings onto various canonical domains, Schiffer’s paper is an early example of using the mappings as a tool in geometric function theory. Moreover, his study of the extremal problem is clearly influenced by his development of variational methods.

Let D be a finitely connected domain containing ∞ in its interior and consider univalent functions F in D with

$$\displaystyle{ F(z) = z + \frac{A} {z} + \frac{B} {{z}^{2}} + \cdots }$$

(1)

near ∞. Let $p(z) = z + a/z + \cdots$ and $q(z) = z + b/z + \cdots$ be the unique conformal mappings of D onto a domain bounded by horizontal slits and vertical slits, respectively. Then

$$\displaystyle{\mathrm{Re}\ b \leq \mathrm{ Re}\ A \leq \mathrm{ Re}\ a.}$$

Thus Re A can vary in an interval of length Re{a − b}, and Schiffer calls this length the span of D. It is the same for any domain that is the image of D under a mapping of the form (1), so in that sense it is a conformal invariant of D. Schiffer proves, among other basic results, that the coefficients A in (1) cover a disk with diameter equal to the span of D.

Now consider conformal mappings of D by functions as in (1) and ask: How large can the area of the complement be? The answer is π∕2 times the span of D. Also, $\Phi (z) = (1/2)(p(z) + q(z))$ is the conformal mapping of D onto the extremal domain, and Φ(D) is bounded by convex curves. Schiffer’s proof is variational, in the course of which he derives some striking identities for the slit mappings of the extremal domain; see [K] for later work on these. Apparently unbeknownst to Schiffer at the time, the convexity property of Φ had already been discovered by Grunsky in his dissertation [G]. But convexity of the boundary curves does not guarantee univalence, a key property that was shown by Schiffer as part of his analysis. This point is also discussed in [] as an independent observation and without variational methods.

An alternate approach to the area problem, once one knows that Φ is univalent and turns out to be the extremal mapping, uses an expression for Φ in terms of kernel functions and harmonic measures, no less striking, and may be found in []. Between [14] and [34], Schiffer revisited the area problem in [22] as an application of orthonormal families to conformal mappings, then quite new. The correct upper bound for the omitted area emerges easily, but the analysis of Φ as the extremal mapping is troublesome, and the full range of representations via kernel functions was not yet realized. Nevertheless, this was a new take on such problems and was influential in subsequent papers. For example, similar area problems were considered by Garabedian and Schiffer in [26], no doubt motivated by [14] and the ideas in [22]. See [N] as well for a compact exposition. Kühnau [Ku] introduced a notion of the span, together with an associated area problem, using quasiconformal mappings onto (inclined) parallel slit domains. He assumed that the complex dilatation of the mapping is identically zero near ∞, thus allowing for a local expansion analogous to (1).

In [AB], analytic and geometric characterizations of the span were reconsidered by Ahlfors and Beurling as examples of their general approach to defining conformal invariants and associated null-sets. Briefly, they define (relative) conformal invariants by forming $M_{\mathfrak{F}}(z_{0},D) =\sup _{\mathfrak{F}}\vert f^{\prime}(z_{0})\vert$, where z ₀ is a fixed point in D and f varies in a class $\mathfrak{F}(D)$ of analytic functions in D that is invariant under conformal mappings of D.¹ If $M_{\mathfrak{F}}(z_{0},D) = 0$ (which generally implies that it vanishes identically), then the complement of D is a null-set for the class $\mathfrak{F}(D)$.

For an analytic approach to the span, consider the class $\mathfrak{D}(D)$ of analytic functions with a fixed bound on the Dirichlet integral, specifically

$$\displaystyle{\iint _{D}\vert f^{\prime}(z){\vert }^{2}\,dxdy \leq \pi.}$$

Then $M_{\mathfrak{D}}{(z_{0},D)}^{2} = (1/2)\mathrm{span}(D)$, and the extremal is given in terms of the slit mappings as $(p - q)/\sqrt{2\,\mathrm{span }(D)}$. For a geometric characterization, the authors use omitted area to define a class of competing functions. Namely, let $\mathfrak{S}\mathfrak{E}(D)$ be the set of univalent functions in D such that $1/(f(z) - f(z_{0}))$ omits a set of area at least π. Then one has again

$$\displaystyle{M_{\mathfrak{S}\mathfrak{E}}{(z_{0},D)}^{2} = (1/2)\mathrm{span}(D),}$$

and the extremal is $(p + q)/$ $\sqrt{ 2\,\mathrm{span}(D)}$; the paper includes a proof that p + q is univalent.

Since $M_{\mathfrak{D}}(z_{0},D) = M_{\mathfrak{S}\mathfrak{E}}(z_{0},D)$, these results serve to describe the identical classes of null-sets for $\mathfrak{D}(D)$ and $\mathfrak{S}\mathfrak{E}(D)$. This direction of research demonstrates the lasting influence of the span in studying small boundaries and removable point sets for classes of analytic functions on plane domains, with similar applications to Riemann surfaces. The latter is tied up with the classification problem for Riemann surfaces, for example, assessing the size of the boundary for the purpose of supporting nonconstant analytic or harmonic functions with a finite Dirichlet integral, necessary for existence theorems. Schiffer himself contributed to this in [85], 22 years after [14]. For a technical discussion see [AS, RS, SN, SO]. For an informal and personal account, see [A].