MATHEMATICS

S. A. GELFER

ON THE MAXIMUM OF THE CONFORMAL RADIUS OF A FUNDAMENTAL DOMAIN OF A DOUBLY PERIODIC GROUP

(Presented by Academician V. I. Smirnov on 29 XI 1956)

Let \(\{D\}\) be a family of simply connected domains \(D\) of the \(w\)-plane, containing the point \(w=0\) and having the following properties: 1) the domain \(D\) contains no points congruent with respect to the group \(T_n\) of transformations \(w' = w + n_1\omega_1 + n_2\omega_2\), where \(\omega_1\) and \(\omega_2\) are constants whose ratio is not real, while \(n_1\) and \(n_2\) are arbitrary integers; 2) the domain \(D\) contains none of the given system of finitely many points \(a_1,\ldots,a_m\) and no points congruent to them with respect to the group \(T_n\).

Among all domains \(\{D\}\) determine the domain that has the greatest conformal radius.

Denote by \(S_a(\omega_1,\omega_2)\) the class of functions

\[ w=f(z)=\sum_{n=1}^{\infty} c_n z^n, \tag{1} \]

regular in the disk \(|z|<1\) and univalently mapping this disk onto domains of the family \(\{D\}\). The stated problem reduces to determining the maximum of \(|f'(0)|=|c_1|\) in the class \(S_a(\omega_1,\omega_2)\).

Theorem 1. If the function \(w=f(z)\in S_a(\omega_1,\omega_2)\) gives the maximum of the functional \(|f'(0)|\), then it satisfies the differential equation

\[ \frac{1}{z^2 w'^2} = A_0+\sum_{i=1}^{m} A_i \zeta(w-a_i)+A_{m+1}\zeta(w)+\wp(w) \tag{2} \]

where \(A_i\) \((i=0,1,\ldots,m+1)\) are constants, with

\[ \sum_{i=1}^{m+1} A_i=0; \]

\(\zeta(w)\), \(\wp(w)\) are the Weierstrass functions constructed on the periods \(\omega_1\) and \(\omega_2\), and it maps the disk \(|z|<1\) onto a domain \(D\) having the following properties:

1) The domain \(D\), containing the point \(w=0\), is a simply connected fundamental domain \(S_0\) of the group \(T_n\) with cuts. Its boundary consists of a finite number of analytic arcs, pairwise congruent with respect to the group \(T_n\), and of piecewise-analytic cuts issuing from the boundary of \(S_0\) and ending at the points \(a_i\) \((i=1,\ldots,m)\) or at points congruent to them.

2) To each pair of congruent arcs of the boundary \(S_0\) and to the simple arcs of the cuts there correspond, on the circle \(|z|=1\), by means of \(w=f(z)\), two arcs of equal length.

Proof. We first derive a variational formula for functions of the class \(S_a(\omega_1,\omega_2)\).

Consider the function

\[ w^*=\Phi(w,|h|)=w+h\,[q_1(w)-q_1(0)]\prod_{i=1}^{m}\frac{\wp(w)-\wp(a_i)}{\wp(w)-b_i}, \tag{3} \]

where \(b_i\) are arbitrary constants; \(q_1(w)=\zeta(w-w_{2m+1})-\zeta(w-w_{2m+2})\), where \(w_{2m+1}\) and \(w_{2m+2}\) are distinct from the roots \(w_k\) \((k=1,\ldots,2m)\) of the equations \(\wp(w)=b_i\) \((i=1,\ldots,m)\), located in the fundamental parallelogram. As in \((^1)\), it is shown that if \(f(z)\in S_a(\omega_1,\omega_2)\) and the image of the disk \(|z|<1\) under the mapping \(w=f(z)\) covers all the points \(w_k\) \((k=1,\ldots,2m+2)\), then the function \(F(z,|h|)=\Phi(f(z),|h|)\), for sufficiently small \(|h|\), will be regular and univalent in some annulus \(r<|z|<1\).

Applying the theorem of G. M. Goluzin \((^2\), Theorem 1), we obtain a function \(f^*(z)\in S_a(\omega_1,\omega_2)\):

\[ \begin{aligned} f^*(z)=&\, f(z)+h[q_1(f(z))-q_1(0)] \prod_{i=1}^{m}\frac{\wp(f(z))-\wp(a_i)}{\wp(f(z))-b_i} \\ &-hz f'(z)\sum_{k=1}^{2m+2}\frac{\beta_k}{z-z_k} +\bar h z^2 f'(z)\sum_{k=1}^{2m+2}\frac{\bar\beta_k}{1-\bar z_k z} +O(|h|^2), \end{aligned} \tag{4} \]

where

\[ \beta_k= \begin{cases} \displaystyle \frac{(-1)^{k+1}}{z_k f'^2(z_k)} \prod_{i=1}^{m} \frac{\wp(f(z_k))-\wp(a_i)}{\wp(f(z_k))-b_i}, & w_k=f(z_k)\quad (k=2m+1,\,2m+2); \\[2.2ex] \displaystyle \frac{q_1(f(z_k))-q_1(0)} {z_k f'^2(z_k)\wp'(f(z_k))} \, \frac{\displaystyle\prod_{i=1}^{m}\bigl(\wp(f(z_k))-\wp(a_i)\bigr)} {\displaystyle\prod_{\substack{i=1\\(\wp(f(z_k))\ne b_i)}}^{m}\bigl(\wp(f(z_k))-b_i\bigr)}, & (k=1,\ldots,2m),\ |z_k|<1. \end{cases} \]

Next, let \(w=f(z)\) be one of the extremal functions, and let \(D\) be the corresponding extremal domain. From the Lindelöf principle it follows that \(D\) is a certain simply connected fundamental domain \(S_0\) of the group \(T_n\), containing the point \(w=0\), with cuts running from the points \(a_1,\ldots,a_m\) to the boundary of \(S_0\). Applying to the function \(f(z)\) the variational formula (4) and taking into account the arbitrariness of \(\arg h\), we obtain from the extremality condition for \(f(z)\) the equality

\[ q_1'(0)+\sum_{k=1}^{2m+2}\frac{\beta_k}{z_k}=0. \tag{5} \]

Fixing \(z_1,\ldots,z_{2m},z_{2m+2}\) and regarding \(z_{2m+1}\) as the variable \(z\), we obtain an equation from which we conclude that the expression \(\frac{1}{z^2 w'^2}\) is an elliptic function of the variable \(w\). Using the known expression of elliptic functions in terms of the Weierstrass functions \(\wp(w)\) and \(\zeta(w)\), we arrive at the differential equation

\[ \frac{1}{z^2 w'^2} = A_0+\sum_{i=1}^{m}\bigl[A_i\zeta(w-a_i)+B_i\zeta(w+a_i)\bigr] +A_{m+1}\zeta(w)+\wp(w). \tag{6} \]

It is not difficult to show that all the coefficients \(B_i=0\).

Thus, equation (2) is established.

Assertions 1) and 2) of the theorem follow from equation (2).

Slightly modifying the proof of M. A. Lavrent’ev’s uniqueness theorem \((^3)\), Theorem 4; see also \((^4)\), p. 66), we obtain:

Theorem 2. A domain \(D\) possessing properties 1) and 2) of Theorem 1 is unique.

Relying on this theorem, in some particular cases one can find extremal domains by conjecture.

With the aid of the differential equation (2) the following theorem is proved.

Theorem 3. The interior angles of the extremal domain \(D\), situated at congruent vertices, are equal.

Consider the net obtained as a result of tiling the plane \(w\) by the extremal domain \(D\) and by the domains congruent to it with respect to the group \(T_n\). From Theorem 3 it follows:

Corollary. The angles between two neighboring arcs issuing from a vertex of the net are equal.

Fig. 1

In what follows we restrict ourselves to the case of real invariants \(g_2\) and \(g_3\) of the function \(\wp(w)\).

Put \(\Delta=g_2^3-27g_3^2\). Three cases are possible ((5), pp. 143—149):

1) \(\Delta>0\). In this case one period \(\omega_1\) is real, and the other \(\omega_2\) is purely imaginary.

2) \(\Delta<0\). In this case the periods \(\omega_1\) and \(\omega_2\) are complex conjugate numbers.

3) \(\Delta=0\). In this case: a) if \(g_2=0\), then one period is finite and the other is equal to \(\infty\); b) if \(g_2=g_3=0\), then \(\omega_1=\omega_2=\infty\).

In case 3 b) the right-hand side of equation (2) is a rational function of \(w\), for which the point \(w=\infty\) is a zero of multiplicity not less than three. By integration we obtain the Laurent formula ((3), p. 180), and for \(m=1\) the Koebe function.

Case 3 a) reduces to the preceding case by mapping a simply connected vertical strip with slits onto the plane with slits.

For \(\Delta\ne0\) we consider the problem of the maximum of \(|f'(0)|\) for the class \(S(\omega_1,\omega_2)\) of functions \(f(z)\) of the form (1), regular in the disk \(|z|<1\) and mapping it univalently onto domains not containing points congruent with respect to the group \(T_n\). The extremal function in this case satisfies the differential equation

\[ \frac{1}{z^2 w'^2}=A_0+\wp(w), \tag{7} \]

where \(A_0=-\wp(w_0)\), and \(w_0\) is a vertex of the extremal domain \(D\). From (7) it follows that \(c_{2k}=0\) \((k=1,2,\ldots)\), whence we conclude that the points \(\pm \frac12\omega_1\) and \(\pm \frac12\omega_2\) lie on the boundary of \(D\).

If \(\Delta>0\), then by Theorem 2 we conclude that the extremal domain will be a rectangle with center at the point \(w=0\) and with sides parallel to the coordinate axes, of lengths respectively \(\omega_1\) and \(|\omega_2|\).

Finally, if \(\Delta < 0\), then, by integrating equation (7), we find for the inverse function the expression

\[ \ln (\overline{z_0}z)=\int_{w_0}^{w}\sqrt{\wp(w)-\wp(w_0)}\,dw, \tag{8} \]

where \(z_0\) is a point of the circle \(|z|=1\) which is mapped to the vertex \(w_0\), \(w_0=f(z_0)\). To determine the form of the extremal domain, we note that, when the point \(w\) lies on its boundary, the right-hand side of (8) must be equal to a purely imaginary number. This will occur only when, for these same values of \(w\), the argument of the expression under the integral sign on the right-hand side of (8) is equal to \(\pi/2\). In Fig. 1 the boundary of the extremal domain for values

\[ \tau=\arg \frac{\omega_2}{\omega_1}\quad \text{from }0\text{ to }\frac{\pi}{2} \]

is represented schematically by solid lines.

Gorky Civil Engineering Institute
named after V. P. Chkalov

Received
23 IX 1956

CITED LITERATURE

\(^{1}\) S. A. Gelfer, DAN, 98, No. 6, 885 (1954).
\(^{2}\) G. M. Goluzin, Matem. sborn., 19 (61), 2, 203 (1946).
\(^{3}\) M. A. Lavrent’ev, Tr. Fiz.-matem. inst. im. V. A. Steklova, 5, 159 (1934).
\(^{4}\) G. M. Goluzin, Usp. matem. nauk, vol. 6, 26 (1939).
\(^{5}\) Yu. S. Sikorskii, Elements of the Theory of Elliptic Functions with Applications to Mechanics, 1936.