UDC 517.54

MATHEMATICS

A. P. KOPYLOV

ON THE RICHNESS OF THE CLASS OF QUASICONFORMAL MAPPINGS OF DOMAINS IN THREE-DIMENSIONAL EUCLIDEAN SPACE

(Presented by Academician M. A. Lavrent’ev, 19 III 1966)

In the case of plane domains, every homeomorphic image of the unit disk whose boundary consists of more than one point is mapped quasiconformally (with arbitrary measurable and bounded distribution of characteristics) onto the disk. In 3-space, however, there exist homeomorphic images of the unit ball \(|x| < 1\) which are not mapped quasiconformally onto the ball, but are bounded domains. The question arises: how powerful a subset do the quasiconformal images of the unit ball form in the set of all homeomorphic images of the latter. Below we set forth one of the approaches to solving this problem.*

In the space \(\overline{H}\) of all bounded homeomorphic images of the unit ball, situated in the 3-space \(E_3\) of Euclid, introduce the metric

\[ \rho(D_1,D_2)=\max\left\{\sup_{x\in cD_1}\rho_{E_3}(x,cD_2),\ \sup_{x\in cD_2}\rho_{E_3}(x,cD_1)\right\}. \]

Here \(D_1\) and \(D_2\) are domains from the space \(\overline{H}\); \(cD = E_3 - D\); \(\rho_{E_3}(A_1,A_2)\) is the ordinary distance between the sets \(A_1\) and \(A_2\) in \(E_3\); \(x\) is a point of \(E_3\). Let us note some properties of the metric \(\rho\).

1. Relation with the convergence of sequences of domains to a kernel in the sense of Carathéodory.

Lemma 1. Let \(\{D_n\}_{n=0}^{\infty}\) be a sequence of elements of \(\overline{H}\). In order that the following conditions hold: 1) for every point \(x\) of \(D_0\) there exists a section \(\{D_n\}_{n=N}^{\infty}\) of the sequence \(\{D_n\}_{n=0}^{\infty}\), for which \(D_0\) is the kernel with respect to the point \(x\); 2) there does not exist a subsequence of the sequence \(\{D_n\}_{n=0}^{\infty}\) possessing a nondegenerate-to-a-point kernel with respect to some point \(x\) of \(cD_0\), it is necessary and sufficient that there exist a ball \(R'\) intersecting all the domains \(D_0,D_1,\ldots\), and that for every such ball \(R\)

\[ \rho(R\cdot D_n,\ R\cdot D_0)\xrightarrow[n\to\infty]{}0 \]

(where \(\rho(R\cdot D_n,R\cdot D_0)\) is defined in the same way as for elements of \(\overline{H}\)).

Corollary 1. Let \(\{D_n\}_{n=0}^{\infty}\) be a sequence of elements of \(\overline{H}\). If

\[ \rho(D_n,D_0)\xrightarrow[n\to\infty]{}0, \]

then for every point \(x\) of \(D_0\) there exists a section \(\{D_n\}_{n=N(x)}^{\infty}\) of the sequence \(\{D_n\}_{n=0}^{\infty}\), for which \(D_0\) is the kernel with respect to the point \(x\) and which converges to its kernel.

Corollary 2. If \(\{D_n\}_{n=0}^{\infty}\subset \overline{H}\), \(D_n \nearrow D_0\), then

\[ \rho(D_n,D_0)\xrightarrow[n\to\infty]{}0. \]

2. Relation with the property of lower semicontinuity of the Hersch coefficient of quasiconformality of a domain \((^{1,2})\). From (2) and Corollary 1 there follows the assertion:

\[ \text{* We consider only bounded homeomorphic images of the ball.} \]

Lemma 2. If \(\{D_n\}_{n=0}^{\infty} \subset H\) and \(\rho(D_n,D_0)\xrightarrow[n\to\infty]{}0\), then

\[ K(D_0)\leq \lim_{n\to\infty} K(D_n). \]

Theorem. The subset \(Q\) of the metric space \((H,\rho)\), consisting of domains admitting a quasiconformal mapping onto a ball, is an everywhere dense \(F_\sigma\)-set of the first category in this space.

Proof. Category. It suffices to prove that, for every number \(C\geq 1\), the set \(Q_C\) of domains \(D\) in \(H\) such that \(K(D)\leq C\) is nowhere dense in \((H,\rho)\).

Fix a number \(C\), \(C\geq 1\), and consider an arbitrary ball

\[ V=V(D_0,\varepsilon)=\{D:\ D\in H,\ \rho(D,D_0)<\varepsilon\} \]

in the space \((H,\rho)\). Let \(x_0\) be some point of the domain \(D_0\), and let \(|x-x_0|<r\) be a Euclidean ball belonging to the domain \(D_0\) and having center \(x_0\) and the greatest possible radius.

Next let \(a\) be one of the boundary points \(\partial D_0\) of the domain \(D_0\), lying on the sphere \(|x-x_0|=r\). Make a radial slit in the ball \(|x-x_0|<r\), and together with it in the domain \(D_0\), of length \(\min\{r,\varepsilon/2\}\). Denote the resulting domain by \(D_0^\varepsilon\). \(D_0^\varepsilon\) is a homeomorphic image of a ball and has infinite coefficient of quasiconformality. The second assertion follows from the results of Väisälä \((^3)\). It is also clear that \(\rho(D_0,D_0^\varepsilon)=\min\{r,\varepsilon/2\}\). We assert that there exists a ball in the space \((H,\rho)\) with center \(D_0^\varepsilon\) containing no domains from the set \(Q_C\). Assuming the contrary, one could obtain a sequence \(\{D_n\}\) of domains from \(Q_C\) such that \(\rho(D_n,D_0^\varepsilon)\xrightarrow[n\to\infty]{}0\). But then from Lemma 2 there follows the relation

\[ K(D_0^\varepsilon)\leq \lim_{n\to\infty} K(D_n)\leq C, \]

which contradicts the infinitude of \(K(D_0^\varepsilon)\). Therefore there exists a ball \(V(D_0^\varepsilon,\delta)\) in the space \((H,\rho)\), containing no elements of the set \(Q_C\), whose radius \(\delta\) is less than the number \(\varepsilon/2\). The latter means that \(V(D_0^\varepsilon,\delta)\subset V(D_0,\varepsilon)\) and, consequently, \(Q_C\) is nowhere dense in \((H,\rho)\).

Density. Let \(D\) be an arbitrary domain in \(H\) and let \(\varepsilon\) be an arbitrary positive number. Our task is to find, in the ball \(V(D,\varepsilon)\), a domain admitting a quasiconformal mapping onto a ball.

Let \(g\) be a homeomorphism of the ball \(B=\{x:\ |x|<1\}\) onto the domain \(D\). There exists such a piecewise-linear homeomorphism \(h\) of the ball \(B\) in \(E_3\) that \(\rho_{E_3}[g(x),h(x)]<\varepsilon/3\) for all \(x\) in \(B\), and any closed subdomain of the ball is represented as the sum of a finite number of pairwise nonintersecting sets, on each of which \(h\) is a linear transformation \((^{4,5})\).

Since \(g(B_n)\nearrow g(B)=D\), \(B_n=\{x:\ |x|<1-1/n\}\), it follows, by virtue of Corollary 2 of Lemma 1, that there exists a natural number \(n_0\) such that \(\rho[g(B_{n_0}),D]<\varepsilon/3\). One can also calculate, using the inequality \(\rho_{E_3}[g(x),h(x)]<\varepsilon/3\), \(x\in B\), that \(\rho[h(B_{n_0}),g(B_{n_0})]\leq 2\varepsilon/3\). Therefore \(\rho[h(B_{n_0}),D]<\varepsilon\), and by virtue of the properties of the mapping \(h\), the domain \(h(B_{n_0})\) is a quasiconformal image of a ball.

It follows from Lemma 2 that the sets \(Q_C\), \(C\geq 1\), are closed in \((H,\rho)\), and since

\[ Q=\sum_{N=1}^{\infty} Q_N, \]

the theorem is completely proved.

Lviv State University
named after Ivan Franko

Received
23 II 1966

CITED LITERATURE

\(^1\) F. W. Gehring, Bull. Am. Math. Soc., 69, No. 2, 146 (1963).
\(^2\) F. W. Gehring, J. Väisälä, Acta Math., 114, 1 (1965).
\(^3\) J. Väisälä, Ann. Acad. Sci. Fenn., Ser. AI 304, 1 (1961).
\(^4\) E. E. Moise, Ann. Math., 55, 215 (1952).
\(^5\) R. H. Bing, Ann. Math., 69, 37 (1959).