UDC 517.53

P. P. Kufarev, B. P. Kufarev

On Two Metric Methods for Defining a Prime End of a Sequence of Plane Domains

(Presented by Academician M. A. Lavrent’ev on 3 January 1969)

In the work \((^1)\), M. A. Lavrent’ev introduced the concept of relative distance in a plane domain, by means of which the nature of the continuity of a conformal mapping is clearly revealed. This concept makes it possible to introduce metrically the notion of a prime end in the sense of Carathéodory. At the same time S. Mazurkiewicz \((^2)\) metrized the space of prime ends of a simply connected domain \(B\) (without, of course, invoking Riemann’s theorem on the existence of a conformal mapping of \(B\) onto a disk). G. D. Suvorov \((^3)\) defined otherwise and systematically used the metric of prime ends for studying the family \(\widehat{BL}\) of more general mappings.

In the article \((^4)\) G. D. Suvorov defined the prime ends of a sequence of plane simply connected domains \((B_i)\) converging to a kernel \(B_0\). From the point of view of boundary correspondence under conformal, quasiconformal, and certain more general mappings (see \((^{3,5})\)), the set of these prime ends is the natural boundary of the sequence of domains.

Here we introduce in two ways the notion of a prime end of a sequence of domains (not necessarily simply connected), using conformally invariant metrics in the domains \(B_i\). In this case the fact of boundary correspondence by such prime ends under a univalent conformal mapping \(f_i : B_i \to D_i\) of the sequence \((B_i)\) becomes obvious. (On this matter there is our brief communication \((^6)\).)

1. Let the domains \(G\) and \(H\), containing the origin, be bounded in the planes \(R_z\) and \(R_w\), respectively. Below the following notation is used: \(\delta\) is the Euclidean metric; \(\operatorname{Fr} M\) is the boundary of the set \(M\); \(\overline{M}\) is the closure of \(M\) in the plane; \(U_i^t\) is the component containing the point \(z=0\) of the set \(\{z \in B_i : \delta(z, R_z \setminus B_i) > t\}\).

We define the distance between nonempty closed subsets \(M, N \subset \overline{G}\) as follows:

\[ \beta(M,N)=\iint_G |\delta(z,M)-\delta(z,N)|\,dx\,dy. \]

It can be proved that in the case under consideration \(\beta\) is equivalent to the Hausdorff distance \((^7,\ \text{p. }166)\) between the sets \(M\) and \(N\).

Let \(\mathfrak{B}\) be the class of all domains \(B_i \subset G\) covering the disk \(|z|<\varepsilon\). In the notes \((^{8,9})\) it was proved that the space \(\mathfrak{B}\) (with kernel convergence) is metrizable by means of the distance

\[ r(B_1,B_2)=\int_0^\varepsilon \beta(U_1^t,U_2^t)\,dt. \]

As is known (see \((^{10,11})\)), in a domain \(B_i \subset G\) one can establish a metric \(\rho_i(z,\zeta)\), invariant under conformal mappings of \(B_i\) and equivalent to the metric \(\delta(z,\zeta)=|z-\zeta|\), such that, if \(B_i\) is finitely connected, then its completion \(\widetilde{B}_i\) with respect to \(\rho_i\) is identifiable with the space of Carathéodory prime ends of the domain \(B_i\).

In what follows we shall assume that \(\rho_i(z,\zeta)\) are uniformly bounded with respect to \(z,\zeta \in B_i\) and \(B_i \in \mathfrak{B}\); otherwise \(\rho_i\) should be replaced by the metric \(\min(1,\rho_i)\).

2. Denote by \(b_i\) an arbitrary element of the completion \(\dot B_i\) of the domain \(B_i\) with respect to \(\rho_i\), and by \(d_i\) an arbitrary element of \(D_i\). Let

\[ \Sigma=\bigcup_{B_i\in\mathfrak B}\dot B_i . \]

(Obviously, from \(B_i=B_j\) it does not follow that \(b_i=b_j\), but the converse is true.)

Define convergence in the set \(\Sigma\) as follows.

Definition*. A sequence \((b_n)_{n=1,2,\ldots}\subset\Sigma\) converges to \(b_0\in\Sigma\) if: 1) \(B_n\to B_0\) (in \(\mathfrak B\)); 2) for every \(\eta>0\) there exists a point \(z_0\in B_0\) at distance \(\rho_0(z_0,b_0)<\eta\) such that \(\rho_n(z_0,b_n)<\eta\) for \(n>N(z_0)\).

Let, in what follows, \(f_i\) be a schlicht conformal mapping of \(B_i\) onto \(D_i\subset H\), continued to a homeomorphism of \(\dot B_i\) onto \(\dot D_i\), with \(f_i(0)=0\), \(f_i'(0)>0\).

Proposition 1. Let \(B_n\to B_0\) and let \((f_n)_{n=1,2,\ldots}\) converge uniformly inside \(B_0\) to the function \(f_0\) (necessarily mapping \(B_0\) schlichtly onto \(D_0\), see \((^{12})\), p. 230). If, moreover, \(d_n=f_n(b_n)\) and \(d_0=f_0(b_0)\), then the relations \(b_n\to b_0\) and \(d_n\to d_0\) are equivalent.

Since \(D_n\to D_0\) (\((^{12})\), p. 230), this proposition is an obvious consequence of the invariance of the metric \(\rho\).

By a Jordan domain we shall mean a domain bounded by a finite number of Jordan curves having no common points.

If one regards a Jordan domain \(B_0\) as a sequence of coincident domains and takes into account the equivalence of \(\rho_0\) and \(\delta\) for such a domain in \(\dot B_0\), then from the definition of convergence of \(b_n\) to \(b_0\) it easily follows that

Proposition 2. If \(B_0\in\mathfrak B\) is a Jordan domain, then the relations \(b_0^j\to b_0\) and \(|b_0^j-b_0|\to0\) are equivalent (here \(b_0,b_0^j\) are identified with points of the plane).

Proposition 3 (follows from 1 and 2). Let \((f_n)_{n=1,2,\ldots}\) converge uniformly to \(f_0\) inside the bounded Jordan domain \(B_0\) (here \(f_n:B_0\to D_n\)). If \(d_{n_j}=f_{n_j}(b_0^j)\) and \(d_0=f_0(b_0)\), then the relations \(|b_0^j-b_0|\to0\) and \(d_{n_j}\to d_0\) are equivalent.

The theorem of G. D. Suvorova \((^4)\) on the correspondence of boundaries under a conformal mapping of a sequence of simply connected domains allows one to assert that, when \(B_n\to B_0\), the space

\[ \sigma=\bigcup_{n=0}^{\infty}\dot B_n \]

may be identified with the space of prime ends of the sequence of domains \((B_n)\).

3. The following theorem shows that the concept of a prime end of a sequence \((B_n)\) can also be introduced by means of a suitable metrization of the space \(\sigma\subset\Sigma\).

Let

\[ \varphi(z,t;b_i)= \begin{cases} \rho_i(z,b_i), & \text{for } z\in U_i^t,\\ 0, & \text{for } z\notin U_i^t . \end{cases} \]

Theorem. Let \(\mathfrak B_0\) be some equivalence class in \(\mathfrak B\) with respect to normalized schlicht conformal mappings, i.e., for any \(B\in\mathfrak B_0\) there exists a conformal mapping \(f\) onto one and the same domain \(D\), with \(f(0)=0,\ f'(0)>0\). Then the space

\[ \Sigma_0=\bigcup_{B_i\in\mathfrak B_0}\dot B_i \]

is metrizable by means of the distance

\[ \widehat{b_1b_2} = \int_0^{\varepsilon}\!\!\iint_G \left|\varphi(z,t;b_1)-\varphi(z,t;b_2)\right|\,dx\,dy\,dt + r(B_1,B_2). \]

4. In conclusion we shall show how the conformally invariant metric \(\rho\) mentioned in 1 can be defined on the basis of a more general (see \((^{13})\)) definition of extremal length than in \((^{11})\).

* Compare \((^3)\), p. 219, Lemma 9.

We shall call a curve a continuous mapping \(\gamma_n\) of an interval \(I\) (open, half-open, or closed) into a domain \(B\). Let \(\Gamma_n=\gamma_n(I)\). A finite collection \(\gamma=(\gamma_n)_{1\le n\le N}\) of curves in \(B\) is called a generalized curve in \(B\).

We shall call a generalized curve \(\gamma\) a net if the set

\[ \Gamma=\bigcup_{n=1}^{N}\Gamma_n \]

is connected and closed relative to \(B\).

By \(B_0^\Gamma\) we denote the component of connectedness of the set \(B\setminus\Gamma\) relative to the point \(z=0\). Let \(B^\Gamma\) be the union (possibly empty) of all components \(C\) of the set \(B\setminus\Gamma\) having the property: \((\operatorname{Fr} C)\setminus\overline{\Gamma}\) includes some component of \(\operatorname{Fr} B\).

We shall say that a net \(\gamma\) separates the points \(z_1\) and \(z_2\) of \(B\), if: 1) \(z_1z_2\subset B_0^\Gamma\) and 2) \(z_1,z_2\in B^\Gamma\) when \(\overline{\Gamma}\cap\operatorname{Fr} B=\varnothing\). The family of all possible nets \(\gamma\) separating the points \(z_1\) and \(z_2\) will be denoted by \(\gamma_{z_1z_2}\). Let \(\Gamma_{z_1z_2}=(\Gamma)_{\gamma\in\gamma_{z_1z_2}}\).

Lemma. If \(\Gamma'\in\Gamma_{z_1z_2}\), \(\Gamma''\in\Gamma_{z_2z_3}\), but \(\Gamma'\), \(\Gamma''\in\Gamma_{z_1z_3}\), then \(\Gamma'\cap\Gamma''\ne\varnothing\) and \(\Gamma'\cup\Gamma''\in\Gamma_{z_1z_2}\).

We define the metric in \(B\) by the equality:

\[ \rho(z_1,z_2)=\lambda^{1/2}(\gamma_{z_1z_2}), \]

where \(\lambda\) is the extremal length of the family \(\gamma_{z_1z_2}\) \((^{13},{}^{14})\).

It is easy to verify that in a neighborhood of each point \(z_0\in B\setminus\{0\}\) the inequality

\[ \frac{|z-z_0|}{\sqrt{\pi}}\le \rho(z,z_0)\le \sqrt{2\pi}\ln^{-1/2}\frac{a(z_0)}{|z-z_0|},\qquad a(z_0)>0, \tag{1} \]

holds, which implies the equivalence of \(\rho\) and \(\delta\) in \(B\setminus\{0\}\).

If \(D\) is a domain whose boundary consists of a finite number of points or circles without common points, then it is not difficult to see that, for each point \(w_0\in\operatorname{Fr}D\),

\[ |w_1-w_2|/\sqrt{\pi}\le \rho(w_1,w_2)\le \sqrt{2\pi}\ln^{-1/2}a(w_0)/\max_{j=1,2}|w_j-w_0| \tag{2} \]

for sufficiently small \(a(w_0)\) and \(w_1,w_2\in\{w\in D:\ |w-w_0|<b<a(w_0)\}\).

Since \(\rho\) is a conformal invariant, it follows from (1) and (2) that \(\rho(z_1,z_2)\) is consistent with the Carathéodory topology in \(B\setminus\{0\}\) for every finitely connected domain \(B\in\mathfrak{B}\). We note that excluding zero does not violate the validity of the propositions formulated in items 2 and 3.

Replacing the Euclidean metric \(\delta\) by the spherical metric, it is easy to dispense with the condition of uniform boundedness of the domains \(B\in\mathfrak{B}\) with respect to \(\delta\) (see item 1).

Tomsk State University
named after V. V. Kuibyshev

Received
23 XII 1968

CITED LITERATURE

\({}^{1}\) M. A. Lavrent’ev, DAN, 4, No. 5, 207 (1936).
\({}^{2}\) S. Mazurkiewicz, Fund. Math., 6, 272 (1936).
\({}^{3}\) G. D. Suvorov, Families of Plane Topological Mappings, Novosibirsk, 1965.
\({}^{4}\) G. D. Suvorov, Matem. sborn., 33 (75), 1, 73 (1953).
\({}^{5}\) B. P. Kufarev, DAN, 181, No. 2, 282 (1968).
\({}^{6}\) P. P. Kufarev, B. P. Kufarev, Abstracts of Brief Scientific Reports, International Mathematical Congress, Section 4, Moscow, 1966, p. 61.
\({}^{7}\) F. Hausdorff, Set Theory, Moscow–Leningrad, 1937.
\({}^{8}\) B. P. Kufarev, Tr. Tomskogo gos. univ., 169, mathematical series, issue 1, 3 (1963).
\({}^{9}\) B. P. Kufarev, Tr. Tomskogo gos. univ., 189, mathematical series, issue 3, 133 (1965).
\({}^{10}\) A. D. Myshkis, Matem. sborn., 25 (67), 3, 387 (1949).
\({}^{11}\) I. S. Gal, Proc. Nat. Acad. Sci. U.S.A., 45, 1629 (1959).
\({}^{12}\) T. M. Goluzin, Geometric Theory of Functions of a Complex Variable, Moscow, 1966.
\({}^{13}\) E. G. Schlesinger, Am. J. Math., 80, 83 (1958).
\({}^{14}\) L. Ahlfors, A. Beurling, Acta Math., 83, 101 (1950).