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Reports of the Academy of Sciences of the USSR
1960. Volume 130, No. 6
MATHEMATICS
B. V. SHABAT
THE METHOD OF MODULI IN SPACE
(Presented by Academician M. A. Lavrent’ev, XI 5, 1959)
In the geometric theory of functions of a complex variable, the notion of the modulus of topological quadrilaterals and rings is widely used. In the last decade this notion has received a new interpretation in the so-called method of extremal lengths, originating with the memoir of Ahlfors and Beurling \((^1)\). This interpretation makes it possible to transfer the method of moduli to Euclidean spaces of any number of dimensions; for greater clarity we shall restrict ourselves to the case of three-dimensional space.
\(1^\circ\). Definition 1. Family of lines. Let in a spatial domain \(D\) there be given some family \(\{C\}\) of rectifiable curves, and also a metric—a measurable nonnegative function \(\rho=\rho(P)\), where \(P\) is a point of \(D\). We shall call the metric \(\rho\) admissible if, for every curve \(C\) of the family,
\[ \int_C \rho\,ds \geqslant 1. \]
The lower bound, over all admissible \(\rho\),
\[ M=\inf_\rho \int_D \rho^3\,d\omega \]
will be called the modulus of the family \(\{C\}\). A homeomorphic mapping \(P_*=f(P)\) of the domain \(D\) onto \(D_*\) will be called \(Q\)-quasiconformal if it has continuous partial derivatives in \(D\), its Jacobian is everywhere positive, and at each point \(P\in D\) its principal linear part transforms a sphere into an ellipsoid whose ratio
\[ p=\frac{a}{c} \]
of the major semiaxis to the minor one is bounded: \(p(P)\leqslant Q\).
Theorem 1. Under \(Q\)-quasiconformal mappings the modulus of a family of curves may change only within bounded limits depending solely on \(Q\); more precisely, if \(\{C_*\}\) is the image of the family \(\{C\}\), then
\[ M\{C\}/Q^2 \leqslant M\{C_*\}\leqslant Q^2M\{C\}. \tag{1} \]
Indeed, let \(\rho(P)\) be an arbitrary metric admissible for the family \(\{C\}\). On \(D_*\) define the function
\[ \rho_*(P_*)=\rho(P)/\mu(P), \]
where \(P=f^{-1}(P_*)\) and \(\mu=r/a\) is the minimal stretching of \(f\) at the point \(P\) (\(r\) is the radius of the sphere corresponding to the ellipsoid); \(\rho_*\) is, evidently, admissible for \(\{C_*\}\), since \(ds_*\geqslant \mu\,ds\) (\(ds_*\) is the element of length of \(C_*\)) and
\[ \int_{C_*}\rho_*\,ds_* \geqslant \int_C \rho\,ds \geqslant 1. \]
But the Jacobian of the mapping
\[ J=\frac{r^3}{abc}=\frac{\mu^3a^2}{bc}\leqslant \mu^3Q^2 \]
(\(a\geqslant b\geqslant c\) are the semiaxes of the ellipsoid transformed into a sphere), therefore
\[ M\{C_*\}\leqslant \inf_{\rho_*}\int_{D_*}\rho_*^3\,d\omega_* =\inf_\rho \int_D \rho^3 \frac{J}{\mu^3}\,d\omega \leqslant Q^2M\{C\}. \]
Applying the same reasoning to the mapping \(f^{-1}\), also \(Q\)-quasiconformal, we obtain the left-hand side of (1).
Definition 2. Family of surfaces. Let \(\{S\}\) be a family of quadrable surfaces in \(D\); the metric \(\rho=\rho(P)\) will be con-
be admissible if for any surface \(S\) of the family
\[ \int_S \rho^2\,d\sigma \geqslant 1. \]
The modulus of the family \(\{S\}\) will be called
\[ M\{S\}=\inf_\rho \int_D \rho^3\,d\omega, \]
where the lower bound is taken over all admissible \(\rho\).
Theorem 2. If \(\{S_*\}\) is the family corresponding to \(\{S\}\) under a \(Q\)-quasiconformal mapping, then
\[ M\{S\}/Q \leqslant M\{S_*\}\leqslant QM\{S\}. \tag{2} \]
From Theorems 1 and 2 it follows that the moduli of families of curves and surfaces are invariant under conformal mappings.
Remark. Using the results of A. I. Markushevich \((^2)\), one can substantially weaken the function-theoretic restrictions imposed above in the definition of \(Q\)-quasiconformal mappings.
2°. Example 1. \(D\) is a rectangular parallelepiped:
\[ 0<x<\alpha,\qquad 0<y<\beta,\qquad 0<z<\gamma; \]
\(\{C\}\) is the family of all rectifiable curves joining the upper and lower bases. We have
\[ M\{C\}=\alpha\beta/\gamma^2. \tag{3} \]
Indeed, since the length of any \(C\) is not less than \(\gamma\), \(\rho_0=1/\gamma\) is an admissible metric, to which there corresponds the volume \(M=\alpha\beta/\gamma^2\). Let \(\rho\) be an arbitrary admissible metric; by Fubini’s theorem and Hölder’s inequality,
\[ \int_D \rho^3\,d\omega = \int_0^\alpha dx\int_0^\beta dy\int_0^\gamma \rho^3\,dz \geqslant \frac{1}{\gamma^2}\int_0^\alpha dx\int_0^\beta dy \left(\int_0^\gamma \rho\,dz\right)^3 \geqslant \frac{\alpha\beta}{\gamma^2}, \]
for, since the rectilinear segments belong to \(\{C\}\), we have
\[ \int_0^\gamma \rho\,dz \geqslant 1; \]
thus (3) is proved.
Example \(1'\). \(D\) is as in Example 1; \(\{S\}\) is the family of all quadrable surfaces whose boundaries lie on the lateral faces of \(D\). The metric
\[ \rho_0=1/\sqrt{\alpha\beta} \]
is admissible, and the corresponding volume is
\[ M=\gamma/\sqrt{\alpha\beta}. \]
Let \(\rho\) be an arbitrary admissible metric; by Fubini’s theorem and Hölder’s inequality,
\[ \int_D \rho^3\,d\omega = \int_0^\gamma dz \iint_{S_z}\rho^3\,d\sigma \geqslant \frac{1}{\sqrt{\alpha\beta}} \int_0^\gamma \left(\iint_{S_z}\rho^2\,d\sigma\right)^{3/2}dz, \]
where \(S_z\) is the section of \(D\) at height \(z\). Since \(S_z\in\{S\}\), we have
\[ \iint_{S_z}\rho^2\,d\sigma \geqslant 1 \]
and
\[ \int_D \rho^3\,d\omega \geqslant \frac{\gamma}{\sqrt{\alpha\beta}}. \]
Thus:
\[ M\{S\}=\gamma/\sqrt{\alpha\beta}. \tag{3′} \]
Remark. Let \(\{C_1\}\), \(\{C_2\}\), and \(\{C_3\}\) be the families of rectifiable curves joining the corresponding opposite faces of \(D\). From (3) we have
\[ M\{C_1\}M\{C_2\}M\{C_3\}=1. \]
If \(\{S_k\}\) is the family of quadrable surfaces ending on the faces free from the ends of \(\{C_k\}\), then from (3) and (3′) we obtain
\[ M\{C_k\}=1/M^2\{S_k\}. \]
Example 2. \(D\) is a spherical sector:
\[ r_1<r<r_2,\qquad 0\leqslant \varphi<2\pi,\qquad 0<\theta<\theta_0 \]
(\(r,\varphi,\theta\) are polar coordinates); \(\{C\}\) is the family of all rectifiable cur-
curves connecting the spherical boundaries of \(D\). We have
\[ M\{C\}=4\pi\sin^2(\theta_0/2)/\ln^2(r_2/r_1). \tag{4} \]
Indeed, the metric \(\rho_0=1/r\ln(r_2/r_1)\) is admissible, since
\[ \int_C \rho_0\,ds \geq \int_{r_1}^{r_2} \rho_0\,dr = 1 \]
for any \(C\), and the corresponding volume is equal to the right-hand side of (4). Let \(\rho\) be an arbitrary admissible metric,
\[ \int_D \rho^3\,d\omega = \int_0^{2\pi} d\varphi \int_0^{\theta_0} \sin\theta\,d\theta \int_{r_1}^{r_2} \rho^3 r^2\,dr; \]
but since, if
\[ \int_{r_1}^{r_2} \rho\,dr \geq 1, \]
then
\[ \int_{r_1}^{r_2} \rho^3 r^2\,dr \geq \frac{1}{\ln^2(r_2/r_1)} \]
(this is proved by the methods of the calculus of variations), it follows that
\[ \int_D \rho^3\,d\omega \geq \frac{1}{\ln^2(r_2/r_1)} \int_0^{2\pi} d\varphi \int_0^{\theta_0} \sin\theta\,d\theta = \frac{4\pi\sin^2(\theta_0/2)}{\ln^2(r_2/r_1)}, \]
as required.
Example 2′. \(D\) as in Example 2; \(\{S\}\) is the family of all quadrable surfaces whose boundaries lie on the conical part of the boundary of \(D\). We have
\[ M\{S\}=\ln(r_2/r_1)/2\sqrt{\pi}\sin(\theta_0/2). \tag{4′} \]
This is proved by methods analogous to the preceding ones, using Hölder’s inequality.
Remark. From (4) and (4′) it follows that \(M\{C\}=1/M^2\{S\}\).
Example 3. \(D\) is a cylindrical ring: \(r_1<r<r_2,\ 0\leq\varphi<2\pi,\ 0<z<H\) (\(r,\varphi,z\) are cylindrical coordinates); \(\{C\}\) is the family of all rectifiable curves connecting the upper and lower bases of \(D\); \(\{S\}\) is the family of all quadrable surfaces homotopic in \(D\) to the rings \(r_1<r<r_2,\ z=z_0\) \((0<z_0<H)\). We have
\[ M\{C\}=1/M^2\{S\}=\pi(r_2^2-r_1^2)/H^2. \tag{5} \]
Example 4. \(D\) as in Example 3; \(\{C\}\) is the family of all closed rectifiable curves homotopic to the circles \(r=r_0,\ z=z_0\) \((r_1<r_0<r_2,\ 0<z_0<H)\); \(\{S\}\) is the family of all quadrable surfaces homotopic to the rectangles \(\varphi=\varphi_0,\ r_1<r<r_2,\ 0<z<H\). We have
\[ M\{C\}=1/M^2\{S\}=H(r_2-r_1)/4\pi^2 r_1r_2. \tag{6} \]
Example 5. \(D\) as in Example 3; \(\{C\}\) is the family of all rectifiable curves homotopic to the segments \(\varphi=\varphi_0,\ z=z_0,\ r_1<r<r_2\) \((0<z_0<H)\); \(\{S\}\) is the family of all quadrable surfaces homotopic to the cylinders \(r=r_0,\ 0<z<H\) \((r_1<r_0<r_2)\). We have
\[ M\{C\}=1/M^2\{S\}=\pi H/2(\sqrt{r_2}-\sqrt{r_1})^2. \tag{7} \]
3°. General properties. We shall call an admissible metric extremal if, in it, the volume of the domain coincides with the modulus of the family of curves or surfaces under consideration. The following theorem asserts the uniqueness of the extremal metric for a given family.
Theorem 3. The extremal metric of a given family of curves or surfaces is determined uniquely, up to a set of volume measure \(0\).
We shall prove the theorem for the case of a family of curves \(\{C\}\). Let \(\rho_1\) and \(\rho_2\) be two extremal metrics for this family, i.e.
\[
\int_D \rho_1^3\, d\omega=\int_D \rho_2^3\, d\omega=M\{C\}.
\]
The metric \(\rho=(\rho_1+\rho_2)/2\) is obviously admissible for this family and, integrating over \(D\) the identity
\[
\frac12(\rho_1^3+\rho_2^3)=\frac18(\rho_1+\rho_2)^3+\frac38(\rho_1-\rho_2)^2(\rho_1+\rho_2),
\]
we find
\[
\int_D(\rho_1-\rho_2)^2(\rho_1+\rho_2)\,d\omega\leq 0,
\]
whence \(\rho_1=\rho_2\) almost everywhere in \(D\). For the case of a family of surfaces the proof proceeds similarly, if one chooses the admissible metric
\[
\rho=\sqrt{\frac12(\rho_1^2+\rho_2^2)}
\]
and uses the identity
\[
\frac12(\rho_1^3+\rho_2^3)=\rho^3+\frac14(\rho_1-\rho_2)^2
\frac{\rho_1^4+4\rho_1\rho_2\rho^2+\rho_2^4}{\rho_1^3+2\rho^3+\rho_2^3}.
\]
The following two theorems extend the well-known Grötzsch principles from the theory of conformal mappings (see, for example, (3)).
Theorem 4. If two families of curves (or two families of surfaces) with moduli \(M_1, M_2\) lie in disjoint domains \(D_1, D_2\), and a third family, lying in \(D=D_1\cup D_2\), is obtained by the union of the given two, then the modulus of the latter satisfies
\[
M\geq M_1+M_2.
\tag{8}
\]
Let \(\rho\) be an admissible metric for the third family; then \(\rho_k=\rho\) in \(D\), \(\rho_k=0\) in \(D\setminus D_k\), will obviously be admissible for the \(k\)-th family \((k=1,2)\), and (8) follows from the identity
\[
\int_D \rho^3\,d\omega
=
\int_{D_1}\rho_1^3\,d\omega+
\int_{D_2}\rho_2^3\,d\omega.
\]
Corollary. Under an “extension” of a family (of surfaces or curves), its modulus can only increase.
Theorem 5. If the families of curves \(\{C_1\}, \{C_2\}\) are situated in disjoint domains \(D_1, D_2\), and each curve of the family \(\{C\}\) consists of one \(C_1\) and one \(C_2\), then
\[
\frac{1}{\sqrt{M\{C\}}}\geq
\frac{1}{\sqrt{M\{C_1\}}}+
\frac{1}{\sqrt{M\{C_2\}}}.
\tag{9}
\]
For families of surfaces \(\{S_1\}, \{S_2\}\) and \(\{S\}\) having the same properties,
\[
\frac{1}{M^2\{S\}}\geq
\frac{1}{M^2\{S_1\}}+
\frac{1}{M^2\{S_2\}}.
\tag{10}
\]
Let \(\rho_k\) be metrics admissible for the families of curves \(\{C_k\}\); then for any \(\lambda\), \(0<\lambda<1\), the metric \(\rho=\lambda\rho_1\) in \(D_1\), \(\rho=(1-\lambda)\rho_2\) in \(D_2\), is obviously admissible for \(\{C\}\), and by integration we obtain
\[
M\{C\}\leq \lambda^3 M\{C_1\}+(1-\lambda)^3 M\{C_2\}.
\]
Choosing
\[
\lambda=\frac{\sqrt{M\{C_2\}}}{\sqrt{M\{C_1\}}+\sqrt{M\{C_2\}}},
\]
we obtain (9). For families of surfaces the proof is analogous; one need only put
\[
\rho=\sqrt{\lambda}\rho_1 \quad \text{in } D_1,\qquad
\rho=\sqrt{1-\lambda}\rho_2 \quad \text{in } D_2,
\]
and then choose
\[
\lambda=\frac{M^2\{S_2\}}{M^2\{S_1\}+M^2\{S_2\}}.
\]
Corollary. Under a “lengthening” of a family (of curves or surfaces), its modulus can only decrease.
Moscow State University
named after M. V. Lomonosov
Received
30 X 1959
References
- L. V. Ahlfors, A. Beurling, Acta Math., 83, 101 (1950).
- A. I. Markushevich, DAN, 28, 301 (1940).
- L. I. Volkovyskii, Quasiconformal Mappings, Lviv, 1954.