UDC 517.54

MATHEMATICS

A. V. SYCHEV

ON QUASICONFORMAL MAPPINGS IN SPACE

(Presented by Academician M. A. Lavrent'ev on 14 V 1965)

In the present note \(Q\)-quasiconformal mappings \((^{1})\) in three-dimensional Euclidean space are considered.

1. Let \(D\) be a ring domain bounded by the sphere \(|X|=1\) and the segment \([0,r]\), and let \(M(D)\) be its modulus. Then \((^{4})\)

\[ \ln \frac{1}{r} \leqslant M(D) \leqslant \ln \frac{\lambda}{r}, \qquad 4 \leqslant \lambda \leqslant 12.4\ldots \]

Since \(M(D)=2\sqrt{\pi}\,M(\Sigma)\), where \(\Sigma\) is the family of all possible surfaces \(\sigma\) separating the boundary components of \(D\), it follows that

\[ \frac{1}{2\sqrt{\pi}}\ln \frac{1}{r} \leqslant M(\Sigma) \leqslant \frac{1}{2\sqrt{\pi}}\ln \frac{\lambda}{r}. \]

These estimates (with the constant \(\lambda=18\)) were obtained earlier by another method by B. V. Shabat.

Hence, as also in \((^{1})\), it follows:

Theorem 1. Let \(Y=f(X)\), \(X=(x_1,x_2,x_3)\), \(Y=(y_1,y_2,y_3)\), \(f(0)=0\), be an arbitrary \(Q\)-quasiconformal mapping of the ball \(|X|<1\) onto itself.

Then at every point of the ball the estimates

\[ (|X|/\lambda)^Q \leqslant |f(X)| \leqslant \lambda |X|^{1/Q}, \qquad 4 \leqslant \lambda \leqslant 12.4\ldots \tag{1} \]

hold.

We shall suppose that an extremal \(Q\)-quasiconformal mapping of the ball \(|X|<1\) onto itself, giving the maximum distortion \(\rho=\rho(r)\), carries at least one plane section passing through \([0,r]\) into a plane section passing through \([0,\rho]\). Then the distortion under the \(Q\)-quasiconformal mapping \(Y=f(X)\), \(f(0)=0\), of the ball \(|X|<1\) onto itself will not exceed the distortion under the \(Q\)-quasiconformal mapping \(w=\varphi(z)\), \(\varphi(0)=0\), of the disk \(|z|<1\) onto itself:

\[ |\varphi(z)| \leqslant \rho, \]

where \(\rho\) is determined from the relation

\[ \frac{K'(\rho)}{K(\rho)} = \frac{1}{Q}\frac{K'(r)}{K(r)}, \]

in which \(K(r)\) and \(K'(r)\) are, respectively, the complete elliptic integrals of the first kind with moduli \(k=r\) and \(k'=\sqrt{1-r^2}\) \((^{2})\), or, as a consequence of this,

\[ 4^{1-Q}|z|^Q \leqslant |\varphi(z)| \leqslant 4^{1-1/Q}|z|^{1/Q}. \]

On the other hand, the author has constructed an example of a \(Q\)-quasiconformal mapping of the ball \(|X|<1\) onto itself for which

\[ \lim_{r\to 0}\frac{\rho}{r^{1/Q}}=4^{1-1/Q}. \]

The example is based on a mapping obtained by rotating a plane extremal mapping of the disk onto itself. This shows that the constant \(\lambda\) in (1) cannot be replaced by any smaller one than \(4^{1-1/Q}\).

Therefore, under the stated assumption, we have

Theorem 2. Let \(Y=f(X)\), \(f(0)=0\), be an arbitrary \(Q\)-quasiconformal mapping of the ball \(|X|<1\) onto itself.

Then at every point of the ball the estimates hold

\[ 4^{1-Q}|X|^{Q}\le |f(X)|\le 4^{1-1/Q}|X|^{1/Q}, \]

and the constant \(4^{1-1/Q}\) cannot be improved.

1. Let \(Q(D)\) be the lower bound of the numbers \(Q\) for which there exist \(Q\)-quasiconformal mappings of the domain \(D\) onto the ball \(|Y|<1\); if such mappings do not exist, we shall put \(Q(D)=\infty\). Following Gehring \((^5)\), we shall call \(Q(D)\) the coefficient of quasiconformality of the domain \(D\), and any \(Q(D)\)-quasiconformal mapping of the domain \(D\) onto the ball \(|Y|<1\) an extremal mapping.

Lemma. Let the function \(w=\varphi(z)\) map \(Q\)-quasiconformally a domain \(G\), whose boundary contains at least one angular point \(P\) with angle \(\pi\alpha\), \(0\le \alpha\le 2\), onto a domain \(G^*\) with smooth boundary, and suppose there exists a neighborhood \(U\) of the point \(P\) in which

\[ |\Delta w/\Delta z|\ge k \quad \text{for } 0\le \alpha\le 1;\qquad |\Delta w/\Delta z|\le 1/k \quad \text{for } 1<\alpha\le 2, \tag{2} \]

where \(k\) is some constant.

Then \(Q\ge \max(\alpha,1/\alpha)\).

Let \(0\le \alpha\le 1\), and let, for sufficiently small \(\varepsilon>0\), \(\Gamma_\varepsilon\) be the family of all possible curves joining the boundary components of the annulus \(\varepsilon^2<|z-P|<\varepsilon\) in \(U\). Then

\[ M(\Gamma_\varepsilon)=\frac{1}{\pi\alpha}\ln\frac{1}{\varepsilon} +o\left(\ln\frac{1}{\varepsilon}\right). \tag{3} \]

On the other hand, under any mapping with \(Q<1/\alpha\), in view of (2) and (3), Theorem 5, the images of the arcs of the circles \(|z-P|=\varepsilon\), \(|z-P|=\varepsilon^2\) will lie in the annuli
\(k\varepsilon\le |\varphi(z)-P^*|\le K\varepsilon\),
\(k\varepsilon^2\le |\varphi(z)-P^*|\le K\varepsilon^2\),
respectively; \(K=K(Q,k)\).

Hence

\[ M(\Gamma_\varepsilon^*)=\frac{1}{\pi}\ln\frac{1}{\varepsilon} +o\left(\ln\frac{1}{\varepsilon}\right) \tag{4} \]

and from (3), (4)

\[ M(\Gamma_\varepsilon^*)=\alpha M(\Gamma_\varepsilon) +o\left(\ln\frac{1}{\varepsilon}\right), \]

i.e. \(Q\ge 1/\alpha\).

This contradiction proves the assertion of the lemma. The proof in the case \(1<\alpha\le 2\) is analogous.

We shall say that a spatial domain \(D\) contains on its boundary a smooth edge \(AB\) with angle \(\pi\alpha\), \(0\le \alpha\le 2\), if at every point \(P\in AB\) the cross-section normal to \(AB\) has angle \(\pi\alpha\).

Theorem 3. Let the function \(Y=f(X)\) map \(Q\)-quasiconformally a domain \(D\), containing on its boundary a smooth edge \(AB\) with angle \(\pi\alpha\), \(0\le \alpha\le 2\), onto the ball \(|Y|<1\), and suppose that at every point \(P\in AB\) the cross-section normal to \(AB\) is transformed into a smooth surface.

Then \(Q\ge \max(\alpha,1/\alpha)\).

The mapping \(f(X)\) can be extended to a homeomorphism of \(D\cup AB\) onto \(\{|Y|<1\}\cup f(AB)\) \((^6)\). Therefore there is a point \(P\in AB\) at which, in the cross-section normal to \(AB\), (2) is satisfied, and application of the lemma gives the assertion of the theorem.

It follows from this that the assertion of Theorem 3 remains valid if the mapping \(f(X)\) satisfies the smoothness condition only at one of those points \(P \in AB\) at which (2) is fulfilled on the normal sections. In the class of such mappings we have

Corollary. If a domain \(D\) contains on its boundary a smooth edge with angle \(\pi\alpha\), \(0 \leq \alpha \leq 2\), then \(Q(D) \geq \max(\alpha,1/\alpha)\), and for \(\alpha=0\), \(Q(D)=\infty\).

Let \((r,\varphi,x_3)\), \((\rho,\theta,y_3)\) be cylindrical coordinates in the spaces \(X\) and \(Y\), respectively, and let \(D\) be the dihedral angle \(0<\varphi<\pi\alpha\), \(0<\alpha\leq 2\). Since the mapping

\[ \rho=r,\qquad \theta=\varphi/\alpha,\qquad y_3=x_3 \tag{5} \]

transforms \(D\) into a half-space and for it \(Q=\max(\alpha,1/\alpha)\), it is extremal by Theorem 3.

Thus, for a dihedral angle \(\pi\alpha\),

\[ Q(D)=\max(\alpha,1/\alpha). \]

The extremal mapping (5) is not unique; the mappings

\[ \rho=kr,\qquad \theta=\varphi/\alpha,\qquad y_3=\varphi(x_3), \]

where \(\varphi(x_3)\in C^1\), \(\varphi'(x_3)\) lies between \(k\) and \(k/\alpha\), and \(k\) is an arbitrary constant, will also be such.

If, in particular, \(D\) is a quarter-space \((\alpha=1/2)\) or the space with the half-plane \(\{x_1>0,\ x_2=0\}\) removed \((\alpha=2)\), then \(Q(D)=2\).

However, if \(D\) is one-eighth of space (an octant), then the double unfolding is not an extremal mapping. Using orthogonal projection, we see that in this case

\[ Q(D)\leq 2+\sqrt{3}\approx 3.72. \]

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
7 V 1965

REFERENCES

1. B. V. Shabat, DAN, 132, No. 5, 1045 (1960).
2. P. P. Belinskii, DAN, 91, No. 5, 997 (1953).
3. P. P. Belinskii, Siberian Math. J., 1, No. 3, 303 (1960).
4. F. W. Gehring, Trans. Am. Math. Soc., 101, No. 3, 499 (1961).
5. F. W. Gehring, Bull. Am. Math. Soc., 69, No. 2, 146 (1963).
6. J. Väisälä, Ann. Acad. Sci. Fenn., Ser. A, 304 (1961).