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Reports of the Academy of Sciences of the USSR
- Volume 145, No. 1
MATHEMATICS
V. A. ZORICH
ON THE CORRESPONDENCE OF BOUNDARIES UNDER \(Q\)-QUASICONFORMAL MAPPINGS OF A BALL
(Presented by Academician M. A. Lavrent’ev on 19 II 1962)
Here we consider homeomorphic \(Q\)-quasiconformal \((^{1})\) mappings of a ball in \(n\)-dimensional \((n \ge 2)\) Euclidean space \(E^n\), and prove a theorem analogous to Koebe’s theorem \((^{2})\) on the correspondence of attainable boundary points under conformal mappings of the disk onto the plane.
Lemma 1. Let \(\{c\}\) be the family of all curves joining in the ball \(R^n \subset E^n\) nonintersecting sets \(l_1, l_2\); let \(\{P_h\}\) \((0 \le h \le H)\) be a one-parameter family of parallel \((n-1)\)-dimensional sections of the ball \(R^n\), intersecting both sets \(l_1, l_2\) and filling a spherical layer \(V^n\) of height \(H\). Then for the modulus \((^{1})\) \(M\{c\}\) of the family \(\{c\}\) the following lower estimate holds
\[ M\{c\} \ge k(n)\frac{H}{d}, \]
where \(k(n)\) is a constant depending only on \(n\); \(d=\displaystyle\sup_{0\le h\le H}\rho(l_1\cap P_h,l_2\cap P_h)^*\).
For simplicity we shall give the proof for \(n=3\); the general case is essentially no different.
Let \(\eta(p)\ge 0\) be an admissible metric of the family \(\{c\}\) (see \((^{1})\)). Then
\[ \iiint_{R^3}\eta^3\,d\omega \ge \iiint_{V^3}\eta^3\,d\omega = \int_0^H\left[\iint_{P_h}\eta^3\,d\sigma\right]dh. \tag{1} \]
Let us estimate the inner integral on the right-hand side of (1). Take two points
\(p_i=p_i(h)\in l_i\cap P_h\) \((i=1,2)\), and in the section \(P_h\) construct the isosceles right triangle \(\Delta_h\) with hypotenuse \(\overline{p_1p_2}\). The straight line \(l\subset P_h\), orthogonal to the segment \(\overline{p_1p_2}\) and passing through its midpoint, divides \(\Delta_h\) into two triangles \(\Delta_h^1,\Delta_h^2\) symmetric with respect to \(l\), and
\[ \iint_{R_h}\eta^3\,d\sigma \ge \iint_{\Delta_h}\eta^3\,d\sigma = \iint_{\Delta_h^1}\eta^3\,d\sigma + \iint_{\Delta_h^2}\eta^3\,d\sigma . \tag{2} \]
In the section \(P_h\) introduce two systems of polar coordinates \((p_1)\), \((p_2)\) with poles at the points \(p_1\) and \(p_2\), respectively. In the first of these systems we shall measure the polar angle from the direction \(\overrightarrow{p_1p_2}\) counterclockwise, and in the second—from the direction \(\overrightarrow{p_2p_1}\) clockwise. Under these conditions the points of the line \(l\) will have the same coordinates in both systems, and the line \(l\) itself is written in the form \(r=\rho/2\cos\varphi\), \(|\varphi|<\pi/2\), where \(\rho=\rho(p_1,p_2)\).
Let \(\eta_i(r,\varphi)\) be the value of the function \(\eta(p)\) at the point determined by the coordinates \((r,\varphi)\) in the corresponding polar system \((p_i)\) \((i=1,2)\); then
\[ \iint_{\Delta_h^i}\eta^3\,d\sigma = \int_0^{\pi/4} d\varphi \int_0^{\rho/2\cos\varphi} \eta_i^3(r,\varphi)\,r\,dr \qquad (i=1,2). \]
\(*\) By \(\rho(M_1,M_2)\) is denoted the Euclidean distance between the sets \(M_1\) and \(M_2\).
Using Hölder’s inequality, we find
\[ \iint_{\Delta_h^i} \eta^3\,d\sigma \ge \int_0^{\pi/4} \frac{2\cos\varphi\,d\varphi}{4\rho} \left[ \int_0^{\rho/2\cos\varphi} \eta_i(r,\varphi)\,dr \right]^3 \qquad (i=1,2), \]
\[ \iint_{\Delta_h=\Delta_h^1+\Delta_h^2} \eta^3\,d\sigma \ge \frac{\sqrt{2}}{4\rho(p_1,p_2)} \int_0^{\pi/4} \left[ \left( \int_0^{\rho/2\cos\varphi} \eta_1(r,\varphi)\,dr \right)^3 + \left( \int_0^{\rho/2\cos\varphi} \eta_2(r,\varphi)\,dr \right)^3 \right]d\varphi . \tag{3} \]
Since \(\eta(p)\ge 0\) is an admissible metric of the family \(\{c\}\), we have
\[ \int_0^{\rho/2\cos\varphi} \eta_1(r,\varphi)\,dr + \int_0^{\rho/2\cos\varphi} \eta_2(r,\varphi)\,dr \ge 1 . \]
Taking also into account that \(a^n+b^n\ge 2^{(1-n)}(a+b)^n\) for \(a\ge 0,\ b\ge 0\), from (3) we obtain
\[ \iint_{\Delta_h}\eta^3\,d\sigma \ge \pi\sqrt{2}/64\,\rho(p_1(h),p_2(h)). \]
Since \(p_1(h), p_2(h)\) are arbitrary points of intersection \(l_1\cap P_h\), \(l_2\cap P_h\), we conclude that
\[ \iint_{\Delta_h}\eta^3\,d\sigma \ge \frac{\pi\sqrt{2}}{64}\, \frac{1}{\rho(l_1\cap P_h,\ l_2\cap P_h)}, \]
and, returning to (2), and then to (1), we finally find
\[ \iiint_{R^3}\eta^3\,d\omega \ge \frac{\pi\sqrt{2}}{64}\,\frac{H}{d}. \]
Since \(\eta\) is an arbitrary admissible metric, Lemma 1 is proved.
On the basis of Lemma 1 the following is proved.
Lemma 2. If \(\{c\}\) is the family of all possible curves joining in the ball \(R^n\) two connected sets that have a common limit point, then \(M\{c\}=\infty\).
We now introduce two definitions. Let \(d\) be the lower bound of the lengths of all possible curves joining in a domain \(D\) the sets \(A\subset D\) and \(B\subset D\); let \(D', A', B'\) be the images of \(D,A\), and \(B\), respectively, under inversion with respect to some nondegenerate sphere of finite radius; let \(d'\) be the lower bound of the lengths of all possible curves joining in \(D'\) the sets \(A'\subset D'\) and \(B'\subset D'\).
Definition 1. The quantity \(\rho_D(A,B)\), equal to \(d\) when \(d'>0\) and equal to \(0\) when \(d'=0\), will be called the distance between the sets \(A\) and \(B\) relative to the domain \(D\).
Definition 2. Two curves \(\lambda_1\subset D\) and \(\lambda_2\subset D\), going to one and the same point\(^*\) \(p\) of the boundary of the domain \(D\), will be regarded as \(D\)-equivalent if, in every neighborhood \(U_p\) of the point \(p\),
\[ \rho_D(\lambda_1\cap U_p,\lambda_2\cap U_p)=0. \]
On the basis of Lemma 2 one easily obtains:
Lemma 3. Let \(p^*=T(p)\) be a homeomorphic \(Q\)-quasiconformal mapping of the ball \(R^n\) onto a domain \(D^*\), under which connected sets \(A\subset R^n\) and \(B\subset R^n\) pass respectively into the sets \(A^*\subset D^*\) and \(B^*\subset D^*\). If
\[ \rho_{R^n}(A,B)=0, \]
then also \(\rho_{D^*}(A^*,B^*)=0\).
From this lemma it follows immediately that:
Lemma \(3'\). Let \(p^*=T(p)\) be a homeomorphic \(Q\)-quasiconformal mapping of the ball \(R^n\) onto a domain \(D^*\), and suppose that under this mapping \(R^n\)-equi-
\(^*\) As a curve \(\lambda\) we take the homeomorphic image \(\lambda:\ p=p(t)\) of the half-interval \([0\le t<1]\), if the point \(p=p(t)\) as \(t\to 1\) tends to some point \(p_0\), possibly infinitely remote; then we say that “the curve \(\lambda\) goes to the point \(p_0\).”
ivalent curves \(\lambda_1 \subset R^n,\ \lambda_2 \subset R^n\) pass respectively into the curves \(\lambda_1^* \subset D^*,\ \lambda_2^* \subset D^*\). If the mapping \(p^* = T(p)\) has a limit as the point \(p\) approaches the boundary of \(R^n\) along each of the curves \(\lambda_1,\lambda_2\), then the curves \(\lambda_1^*, \lambda_2^*\) are \(D^*\)-equivalent.
Lemma 4. If \(p^* = T(p)\) is a homeomorphic \(Q\)-quasiconformal mapping of the ball \(R^n\), then in \(R^n\) one cannot select a sequence of arcs \(\gamma_m\) whose ends converge to two distinct points of the boundary sphere \(\Gamma\) and whose images \(\gamma_m^* = T(\gamma_m)\) shrink to a point.
Proof. Suppose that, contrary to the assertion of the lemma, it has been possible to select in \(R^n\) a sequence of arcs \(\gamma_m\) whose ends converge to two distinct points \(p_1 \in \Gamma,\ p_2 \in \Gamma\), and whose images \(\gamma_m^* = T(\gamma_m)\) shrink to the point \(p_0^*\). Let \(l\) be a segment of length \(\rho(p_1,p_2)/2\), parallel to the segment \(\overline{p_1p_2}\) and symmetric with respect to the center of the ball \(R^n\). Consider the sequence of families \(\{c_m\}\) of all curves joining in \(R^n\) the segment \(l\) with the corresponding arc \(\gamma_m\). Without loss of generality, the ends of the arcs \(\gamma_m\) may be assumed so close to the points \(p_1,p_2\), respectively, that each of the two \((n-1)\)-dimensional planes drawn orthogonally to the segment \(l\) through its ends intersects all the arcs \(\gamma_m\). Then, by Lemma 1, for any family \(\{c_m\}\) we have:
\[ M\{c_m\} \geqslant k(n)\frac{\rho(p_1,p_2)}{2r}=\varepsilon^{2(n-1)}>0, \]
where \(r\) is the radius of the ball \(R^n\).
Since the mapping \(p^* = T(p)\) is \(Q\)-quasiconformal, the following estimate (1) must hold for the modulus \(M\{c_m^*\}\) of the image of the family \(\{c_m\}\):
\[ M\{c_m^*\} \geqslant Q^{-(n-1)} M\{c_m\} \geqslant \frac{\varepsilon^{2(n-1)}}{Q^{(n-1)}} \qquad (m=1,2,\ldots). \tag{4} \]
Now let \(p_0^*\) be the point*, to which the images \(\gamma_m^*\) of the arcs \(\gamma_m\) shrink, and let \(\rho(p^*,p_0^*) < r_1^*\) and \(\rho(p^*,p_0^*) < r_2^*\) \((0<r_1^*<r_2^*)\) be two concentric balls containing no points of the image of the segment \(l\). Since the arcs \(\gamma_m^*\) shrink to the point \(p_0^*\), for any value \(r_1^*>0\) there is a number \(m(r_1^*)\) such that for all \(m>m(r_1^*)\) the arcs \(\gamma_m^*\) will lie in the ball \(\rho(p^*,p_0^*)<r_1^*\). But then, for \(m>m(r_1^*)\), all the curve families \(\{c_m^*\}\) will intersect both spheres \(\rho(p^*,p_0^*)=r_1^*,\ \rho(p^*,p_0^*)=r_2^*\), and by Grötzsch’s principles (1) the modulus \(M\{c_m^*\}\) of each of these families is not greater than the modulus of the family of curves lying in the layer \(r_1^*\leqslant \rho(p^*,p_0^*)\leqslant r_2^*\) and joining the spherical boundaries of the layer. The modulus of the latter family is equal to \(nV_n/[\ln(r_2^*/r_1^*)]^{n-1}\), where \(V_n\) is the volume of the \(n\)-dimensional unit ball.
Thus,
\[ M\{c_m^*\}\leqslant nV_n/[\ln(r_2^*/r_1^*)]^{\,n-1} \quad \text{for } m>m(r_1^*). \tag{5} \]
Putting in estimate (5)
\[ r_1^*=r_2^*\exp\left[-\frac{(2nV_n)^{1/(n-1)}Q}{\varepsilon^2}\right], \]
we obtain
\[ M\{c_m^*\}\leqslant \frac{\varepsilon^{2(n-1)}}{2Q^{\,n-1}} \quad \text{for } m>m(r_1^*), \]
which contradicts relation (4). Lemma 4 is proved.
Definition 3. The collection \((p,\gamma)\) consisting of a point \(p\) of the boundary of the domain \(D\) and a curve \(\gamma\) lying in \(D\) and going to the point \(p\) is called an attainable boundary point of the domain \(D\). Two attainable boundary points \((p,\gamma_1)\) and \((p_2,\gamma_2)\) of the domain \(D\) are considered coincident if and only if the curves \(\gamma_1\) and \(\gamma_2\) are \(D\)-equivalent.
* Here \(p_0^*\) is regarded as a finite point; if necessary, this can be achieved by an inversion.
It is clear from the definition that an attainable boundary point \((p,\gamma)\) can be identified with a class of \(D\)-equivalent curves going to the point \(p\). Thus the attainable point \((p,\gamma)\), like this entire class, is completely determined by specifying one curve of the class.
Theorem. Under a homeomorphic \(Q\)-quasiconformal mapping of the ball \(R^n\) onto a domain \(D^*\), to each attainable point \((p_0^*,\gamma_0^*)\) of the boundary \(\Gamma^*\) of the domain \(D^*\) one can assign a definite point \(p_0\) of the boundary sphere \(\Gamma\) in such a way that if a point \(p^*\) tends to \(p_0^*\) along any curve \(\gamma_0^*\) defining \((p_0^*,\gamma_0^*)\), then the corresponding point \(p\) tends to \(p_0\), and different attainable points of \(\Gamma^*\) correspond to different points of \(\Gamma\). The set \(E\) of points of the sphere \(\Gamma\) corresponding to all attainable boundary points of the domain \(D^*\) is everywhere dense on every continuum \(K \subset \Gamma\).
Proof. Let \(\gamma_0^* \subset D^*\) be a curve defining the attainable boundary point \((p_0^*,\gamma_0^*)\), and let \(\gamma_0\) be the preimage of this curve under the given \(Q\)-quasiconformal mapping. The curve \(\gamma_0\) has on the sphere \(\Gamma\) only one limit point, for otherwise we could select in the ball \(R^n\) a sequence of arcs with endpoints converging to two distinct boundary points, whose images under a homeomorphic \(Q\)-quasiconformal mapping would contract to a point, which is impossible by Lemma 4. Thus, if a point \(p_0^* \in \gamma_0^*\) moves along \(\gamma_0^*\), approaching \(p_0^*\), then the corresponding point \(p \in \gamma_0\), moving along \(\gamma_0\), approaches one completely determined point \(p_0 \in \Gamma\). We assign this point \(p_0\) to the attainable boundary point \((p_0^*,\gamma_0^*)\).
The established correspondence is correct, i.e., it does not depend on the choice of the curve defining the attainable boundary point. Indeed, if the curves \(\gamma_1^*\) and \(\gamma_2^*\) define one and the same attainable boundary point \((p^*,\gamma^*)\), then in any neighborhood of the point \(p^*\) they can be joined by a curve lying in \(D^*\) and not leaving this neighborhood. If the curves \(\gamma_1,\gamma_2\)—the preimages of the curves \(\gamma_1^*,\gamma_2^*\)—defined different points \(p_1,p_2\) on the sphere \(\Gamma\), then the preimages of the sequence of curves joining \(\gamma_1^*\) with \(\gamma_2^*\) and contracting to the point \(p^*\) would be arcs with endpoints converging to distinct points \(p_1 \in \Gamma,\ p_2 \in \Gamma\), which again contradicts Lemma 4.
If the curves \(\gamma_1^*\) and \(\gamma_2^*\) define different attainable boundary points, then without loss of generality one may assume that \(\rho_{D^*}(\gamma_1^*,\gamma_2^*)>0\). But then from Lemma \(3'\) it follows at once that, under our correspondence, distinct attainable points of the boundary of the domain \(D^*\) correspond to distinct points on the sphere \(\Gamma\).
It remains to show that the set \(E\) of points of the sphere \(\Gamma\) corresponding to all attainable boundary points of the domain \(D^*\) is everywhere dense on every continuum \(K \in \Gamma\). Suppose that a continuum \(K \subset \Gamma\) distinct from a point contains not a single point of the set \(E\). This means that along no curve going to some point of the continuum \(K\) can our mapping have a limit; i.e., the length of the image of any such curve is infinite. But then the modulus of the image \(\{c^*\}=T(\{c\})\) of the family \(\{c\}\) of curves joining in \(R^n\) an arbitrary segment \(\overline{p_1p_2}\subset R^n\) with the continuum \(K\) is equal to zero, whereas the modulus of the family \(\{c\}\) itself, by Lemma 1, is greater than zero. This contradicts the fact that the mapping \(p^* \equiv T(p)\) is \(Q\)-quasiconformal. The theorem is proved.
In conclusion I express my gratitude to B. V. Shabat for his attention and assistance in the work.
Moscow State University
named after M. V. Lomonosov
Received
8 II 1962
References
- B. V. Shabat, DAN, 130, No. 6 (1960).
- P. Koebe, J. reine u. angew. Math., 145, 201 (1915).