Abstract
Full Text
B. V. SHABAT
ON THE THEORY OF QUASICONFORMAL MAPPINGS IN SPACE
(Presented by Academician M. A. Lavrent’ev on 17 II 1960)
Here we consider applications of the method of moduli, extended to space (¹). This method leads to simple proofs of the theorems of M. A. Lavrent’ev (²) and M. A. Kreines (³) on quasiconformal mappings in space, as well as to a certain strengthening of them.
By a quasiconformal mapping of a spatial domain (D) we shall here mean a homeomorphic mapping (P_ = f(P)) of this domain, possessing at each of its points continuous partial derivatives and a positive Jacobian (J). At each point (P \in D), the principal linear part of (f) transforms spheres into similar and similarly situated ellipsoids, which we shall call characteristic; the ratios of the semiaxes (p=a/c) and (q=b/c) ((a \ge b \ge c)) of these ellipsoids will be called the characteristics of the mapping at the point (P). Quasiconformal mappings with characteristics bounded by a constant (Q) are called (Q)-quasiconformal*.
(1^\circ). We begin with a lemma which extends to space the well-known lemma of Grötzsch–Teichmüller (see, for example, (⁴)).
Lemma. Among all families of smooth homeomorphic sphere surfaces lying in the ball (OP<1) and enclosing continua (\gamma) that contain (O) and a fixed point (P_0 \ne O), the greatest modulus is attained by the family of surfaces ({S}) enclosing the rectilinear segment (OP_0).
Let (r,\varphi,z) be cylindrical coordinates (with origin (O), the (z)-axis along (OP_0)); note first of all that, in computing the modulus of the extremal family ({S}), one may restrict oneself to metrics of the form (\rho=\rho(r,z)). Indeed, along with every admissible (\rho(r,\varphi,z)), the metrics (\rho_\alpha(r,\varphi,z)=\rho(r,\varphi+\alpha,z)) ((0<\alpha<2\pi)) are also admissible, for together with (S) the family also contains the surfaces obtained from (S) by rotations about the (z)-axis. But then the metric
[
\rho_0(r,z)=\frac{1}{\sqrt{2\pi}}\left(\int_0^{2\pi}\rho_\alpha^2\,d\alpha\right)^{1/2}
=
\frac{1}{\sqrt{2\pi}}\left(\int_0^{2\pi}\rho^2\,d\varphi\right)^{1/2}
]
is also admissible, since
[
\int_S \rho_0^2\,d\sigma
=
\frac{1}{2\pi}\int_S d\sigma\int_0^{2\pi}\rho_\alpha^2\,d\alpha
=
\frac{1}{2\pi}\int_0^{2\pi} d\alpha\int_S \rho_\alpha^2\,d\sigma
\ge
\frac{1}{2\pi}\int_0^{2\pi} d\alpha
=
1
]
for every (S). But, by Hölder’s inequality, the volume of the ball in the metric (\rho_0),
[
\int_V \rho_0^3\,d\omega
=
\frac{1}{2\pi\sqrt{2\pi}}\int_V d\omega\left(\int_0^{2\pi}\rho_\alpha^2\,d\alpha\right)^{3/2}
\le
\frac{1}{2\pi}\int_V d\omega\int_0^{2\pi}\rho_\alpha^3\,d\alpha
=
\frac{1}{2\pi}\int_0^{2\pi} d\alpha\int_V \rho_\alpha^3\,d\omega
=
\int_V \rho^3\,d\omega
]
does not exceed the volume of this ball in the metric (\rho).
It is also not difficult to see that (M{S}=M{\widetilde S}), where ({\widetilde S}) is the family of all smooth surfaces, homeomorphic to a sphere, of revolution about the (z)-axis and enclosing (OP_0).
Now let ({\Sigma}) be a family of smooth surfaces, homeomorphic to a sphere, enclosing the continuum (\gamma). Denote by (\widetilde\Sigma) the surface obtained from (\Sigma) by the following symmetrization process: in each plane (z=z_0) one constructs a circle with center on the (z)-axis whose area (\pi r_0^2) is equal to the area of the section, by this plane, of the domain bounded by (\Sigma); (\widetilde\Sigma) is formed by the circumferences of such circles. The family ({\Sigma}) passes into the family of all surfaces of revolution ({\widetilde\Sigma}) enclosing the continuum (\widetilde\gamma). Since (\widetilde\gamma \supset OP_0), it follows, by Theorem 4 of ((^1)), that
[
M{\widetilde\Sigma}\leq M{\widetilde S}=M{S}.
]
Denote by ({\Sigma_t}) an arbitrary one-parameter family of surfaces from ({\Sigma}) simply covering the ball with the continuum (\gamma) removed; let ({\widetilde\Sigma_t}) be the family obtained by its symmetrization; it simply covers the ball with (\widetilde\gamma) removed. Let (\widetilde\rho(r,z)) be an arbitrary admissible metric for ({\widetilde\Sigma_t}); define the metric (\rho) by setting it equal to (\widetilde\rho(r_0,z_0)) at all points of the section of (\Sigma_t) by the plane (z=z_0). The metric (\rho) is admissible for ({\Sigma_t}), since symmetrization does not increase surfaces, and, since symmetrization does not change volumes,
[
\int_{V\setminus\gamma}\rho^3\,d\omega=\int_{V\setminus\widetilde\gamma}\widetilde\rho^{\,3}\,d\omega.
]
Hence it follows that
[
M{\Sigma_t}\leq M{\widetilde\Sigma_t}\leq M{\widetilde\Sigma}\leq M{S},
]
and from this it is not hard to derive also the inequality (M{\Sigma}\leq M{S}). The lemma is proved.
By invariance under rotations, the modulus of the extremal family depends only on (r=OP_0); we shall denote it by (\nu=\nu(r)). The function (\nu(r)) is decreasing, (\nu(0)=\infty), (\nu(1)=0). Using the Grötzsch principles from ((^1)) and supplementary conformal self-mappings of the ball, one can obtain for it the following estimates:
[
\frac{1}{2\sqrt{\pi}}\ln\frac{1}{r}<\nu(r)<\frac{1}{2\sqrt{\pi}}\ln\frac{16}{r}.
\tag{1}
]
(2^\circ). The following theorem extends to space the well-known theorem of M. A. Lavrent’ev on distortion ((^5)), in the form given to it by Hersch ((^6)).
Theorem 1. Let (P_ = f(P)), (f(O)=O), be an arbitrary (Q)-quasiconformal mapping of the ball (\overline{OP}<1) onto itself; for any point of the ball we have
[
\frac{1}{Q}\nu(\overline{OP})\leq \nu(\overline{OP_})\leq Q\nu(\overline{OP}).
\tag{2}
]
The proof is based on the lemma of (1^\circ) and Theorem 1 of ((^1)). Using the monotonicity of (\nu(r)), one may rewrite (2) in the form
[
\nu^{-1}{Q\nu(\overline{OP})}\leq \overline{OP_*}\leq
\nu^{-1}\left{\frac{1}{Q}\nu(\overline{OP})\right}.
\tag{3}
]
Corollary 1 (M. Ried ((^7))). A conformal mapping of the ball (\overline{OP}<1) onto itself, normalized by (f(O)=O), reduces to a rotation.
This is obtained from (3) for (Q=1).
Corollary 2. Under the hypotheses of Theorem 1,
[
\left(\frac{\overline{OP}}{16}\right)^Q\leq \overline{OP_*}\leq 16(\overline{OP})^{1/Q}.
\tag{4}
]
This estimate substantially sharpens the estimate of M. A. Kreines ((^3)); as the example of the mapping (r_=r^{1/Q}), (\theta_=\theta), (\varphi_*=\varphi) ((r,\varphi,\theta) are polar coordinates) shows, its order is sharp.
Remark. If one applies a slightly modified method of continuation by symmetry (5), then estimate (4) can be extended to arbitrary points (P_1) and (P_2) of the ball (\overline{OP}\leqslant 1):
[
(\overline{P_1P_2}/K)^Q \leqslant \overline{P_{1}P_{2}} \leqslant K(\overline{P_1P_2})^{1/Q},
\tag{5}
]
where (K\leqslant 17\cdot 32^Q).
3°. The method under consideration leads to a simple proof of a theorem of M. A. Lavrent'ev (2); extending the method of K. Andreian-Cazacu (8) to space makes it possible to strengthen the result. Suppose that in (D) there is given a family ({S}) of smooth surfaces such that through every point (P\in D) there passes one and only one (S), and a quasiconformal mapping (P_*=f(P)). Let (\alpha_1,\alpha_2,\alpha_3) denote the angles formed by the normal to (S) at (P) with the semiaxes of the characteristic ellipsoid. The element of area of (S) cut out by this ellipsoid is equal to
[
d\sigma=\frac{\pi abc}{\sqrt{a^2\cos^2\alpha_1+b^2\cos^2\alpha_2+c^2\cos^2\alpha_3}}.
]
Let (r) be the radius of the sphere corresponding to the ellipsoid; then the corresponding area element is (d\sigma_*=\pi r^2), and the Jacobian is (J=r^3/abc); therefore
[
\frac{d\sigma_*}{d\sigma}
=
J^{2/3}
\left{
\left(\frac{p^2}{q}\right)^{2/3}\cos^2\alpha_1
+
\left(\frac{q^2}{p}\right)^{2/3}\cos^2\alpha_2
+
\left(\frac{1}{pq}\right)^{2/3}\cos^2\alpha_3
\right}^{1/2}
=
J^{2/3}{m(P)}^{2/3}.
\tag{6}
]
Let (\rho(P)) be an admissible metric for ({S}); for the corresponding family ({S_}) put
[
\rho_(P_)=J^{-1/3}{m(P)}^{-1/3}\rho(P).
]
Since by (6) (\rho_^2\,d\sigma_=\rho^2\,d\sigma), this metric is admissible for ({S_}). The volume of (D_*) in this metric is
[
\int_{D_}\rho_^3\,d\omega_*
=
\int_D \frac{\rho^3(P)}{m(P)}\,d\omega.
]
Suppose further that the surfaces (S=S_t) depend on a parameter (t), (t_1<t<t_2), with (d\omega=d\sigma_t\,dt) ((d\sigma_t) is the area element of (S_t)), and denote
[
m(t)=\max_{P\in S_t} m(P).
]
Using Hölder’s inequality, we find
[
\int_{D_}\rho_^3\,d\omega_*
\geqslant
\int_{t_1}^{t_2}\frac{dt}{m(t)}\int_{S_t}\rho^3\,d\sigma_t
\geqslant
\int_{t_1}^{t_2}\frac{dt}{m(t)}\,\frac{1}{\sqrt{\sigma(t)}}
\left(\int_{S_t}\rho^2\,d\sigma_t\right)^{3/2},
]
where (\sigma(t)) is the area of (S_t). Since (\rho) is admissible for ({S_t}), finally
[
\int_{D_}\rho_^3\,d\omega_*
\geqslant
\int_{t_1}^{t_2}\frac{dt}{m(t)\sqrt{\sigma(t)}}.
\tag{7}
]
On the basis of these considerations one proves
Theorem 2. Let (P_=f(P)) be a quasiconformal mapping of the unit ball with the center removed onto itself, mapping the unit sphere onto itself. Denote by (\alpha_i) the angles formed by (OP) with the semiaxes of the characteristic ellipsoid and
[
m(r)=\max_{\overline{OP}=r} m(P),\qquad 0<r<1.
]
If*
[
\int_0^1\frac{dr}{r\,m(r)}=\infty,
\tag{8}
]
then there exists (\displaystyle \lim_{P\to O} f(P)).
Indeed, suppose that the set of limit values of (f(P)) at the point (O) is a continuum (\gamma), not a point; by subjecting the ball to an additional conformal mapping, one can arrange that (\gamma) contains (O).
By the lemma from (1^\circ), the modulus of the family of surfaces surrounding (\gamma) is finite. On the other hand, for any admissible metric (\rho_*), according to (7),
[
\int_{D_}\rho_^3\,d\omega_*\geq
\frac{1}{2\sqrt{\pi}}\int_0^1\frac{dr}{r m(r)}=\infty .
]
Remark. Let
[
p(r)=\max_{\overline{OP}=r} p(P);
]
obviously, (m(r)\leq p(r)), and condition (8) is satisfied if
[
\int_0^1 \frac{dr}{r p(r)}=\infty .
]
This condition is certainly satisfied for (Q)-quasiconformal mappings.
(4^\circ). The following two theorems are special cases of a theorem of M. A. Lavrent'ev ((^2)).
Theorem 3. There does not exist a (Q)-quasiconformal mapping of the half-space (z>0) with the removed segment (l={x=y=0,\ 00) ((x,y,z) are Cartesian coordinates).
If (l) is transformed into a point, let it be (O), i.e., suppose that there exists (\lim f(P)=O) as (P\to l). Then we consider a cylinder of radius (R) and height (H), having (l) as its axis, and cover it by a hemisphere of the same radius with center at the upper end of (l). Let ({S}) be the family of all quadrable surfaces lying inside this surface and surrounding (l), whose boundaries lie on the lower base of the cylinder; let (S_1) and (S_2) be the parts of these surfaces lying respectively in the cylinder and in the hemisphere. By Theorem 5 and formulas (10) and (13) of ((^1)), the moduli of these families are connected by the relation
[
\frac{1}{M^2{S}}\geq \frac{1}{M^2{S_1}}+\frac{1}{M^2{S_2}}=\frac{\pi H}{2R},
]
and, consequently, (M{S}) is finite. At the same time, for the corresponding family (M{S_*}=\infty); the contradiction excludes this case.
If (l) is not transformed into a point, then the set of indeterminacy for (l), i.e., the totality of all limit values (f(P)) over all possible sequences tending to points of (l), is a continuum distinct from a point and lying in the plane (z_=0). From topological considerations it is clear that if, for some point (P_0\in l), the set of indeterminacy degenerates into a point (P_), then for the entire segment from (P_0) to the upper end of (l) this set coincides with (P_*). Therefore, in the case under consideration there exists a segment (OP_0\subset l) such that, for all its points, the set of indeterminacy is a continuum distinct from a point. Consider a cylinder of radius (R) and height (h=OP_0), with the axis (OP_0) removed, and in it the family of all closed rectifiable curves surrounding the axis. By formula (12) of ((^1)), the modulus of this family is infinite, but corresponding to it is a family of curves whose lengths are bounded below by a positive constant and which therefore has finite modulus. The theorem is proved.
By analogous methods one proves
Theorem 4. There does not exist a (Q)-quasiconformal mapping of the half-space (z>0) with the removed piece of the plane (\Pi={x=0,\ -R\leq y\leq R,\ 00).
Moscow State University
named after M. V. Lomonosov
Received
18 I 1960
References
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- M. A. Lavrent'ev, DAN, 20, No. 4, 241 (1938).
- M. A. Kreines, Mat. sborn., 9, 3, 713 (1941).
- L. I. Vol'kovyskii, Quasiconformal mappings, L'vov, 1954.
- M. A. Lavrent'ev, Mat. sborn., 42, 407 (1935).
- J. Hersch, Comm. Math. Helv., 30, No. 1 (1956).
- M. Reade, Mich. Math. J., 4, No. 1, 64 (1957).
- K. Andreian-Cazacu, DAN, 126, No. 2 (1959).