MATHEMATICS

D. B. POTYAGAILO

ON THE PROBLEM OF GLUING TWO HALF-DISKS

(Presented by Academician M. A. Lavrent’ev on 30 V 1961)

In the present note a class of Riemann surfaces is indicated, the determination of whose type is directly reduced to the determination of the type of the gluing of two half-disks for a given gluing function from a sufficiently broad class.

1. Let \(\bar{x}=\varphi(x)\) be a topological correspondence between the points of the boundary diameters of the half-disks \(D_1(\operatorname{Im} z\leqslant 0,\ |z|<\rho)\) and \(D_2(\operatorname{Im} z\geqslant 0,\ |z|<\rho)\), \(\varphi(\pm\rho)=\pm\rho,\ \rho\leqslant\infty\), such that there exists a pair of analytic functions \(\hat{z}=f_1(z)\) and \(\hat{z}=f_2(z)\), regular and univalent in \(D_1\) and \(D_2\), which transform the latter into domains filling the disk \(|\hat{z}|<R\leqslant\infty\), while \(f_1[\varphi(x)]=f_2(x)\) for all \(x\in(-\rho,\rho)\). For convenience, instead of \(D_1,D_2\) we consider the half-strips \(B_r,\ r=1,2:\ (-1)^r\operatorname{Im} z\geqslant0,\ |\operatorname{Re}z|<\rho\), with the same correspondence between the points of adjoining bases. The passage from the problem of gluing \(D_r\) to the problem of gluing \(B_r\) with gluing function \(\varphi(x)\) is ensured by an elementary \(q\)-quasiconformal transformation. The half-strips \(B_r\) with gluing function \(\varphi(x)\) are understood as a connected univalent domain and are denoted by \(\Phi\) \((\varphi,\rho\leqslant\infty)\). If \(F\) is an arbitrary simply connected Riemann surface, and \(z=f(w)\) is a function mapping \(F\) conformally onto the disk \(|z|<R\leqslant\infty\), then the distribution of the characteristics \(p(w),\theta(w)\) on \(F\) preserves the type in the case when there exists a quasiconformal mapping with characteristics \(p(z),\theta(z)\) of the disk \(|z|<R\) onto the disk \(|\hat{z}|<R\) \((^1)\).

Let \(\{a_{rk}\},\ r=1,2,\ a_{rk}<a_{rk+1},\ k=0,\pm1,\pm2,\ldots\), be arbitrary sequences of real numbers, \(\lim a_{rk}=\pm\infty\) as \(k\to\pm\infty\) and \(r=1,2\). In each half-strip \(S_{rk}: u\in(a_{rk-1},a_{rk}),\ (-1)^r v>0\), on a copy of the \(w\)-plane, \(w=u+iv\), on the sheet \(H\) make cuts along the segments

\[ w=u+i(-1)^r\frac{2n-1}{m_{rk}},\quad n=1,2,\ldots,\quad a_{rk-1}+m_{rk}^{-1}< \]

\[ <u<a_{rk}-m_{rk}^{-1}, \]

where \(\{m_{rk}\}\) are two sequences of real numbers,

\[ m_{rk}>2(a_{rk}-a_{rk-1})^{-1}=2\Delta_{rk}^{-1} \]

for all \(k\) and \(r\). To each cut we glue a copy of the \(w\)-plane—the sheet \(H_{nk}^r\) with an analogous cut. The open simply connected Riemann surface constructed in this way has, over each disk \(|w|<R\), only algebraic branch points of the first order. Denote by \(\mathfrak{A}\) the class of surfaces of this kind obtained under all possible choices of the sequences \(\{a_{rk}\}\) and \(\{m_{rk}\}\). The rectangle

\[ R_{nk}^r:\ u\in(a_{rk-1},a_{rk}),\quad v\in\left((-1)^r\frac{2n}{m_{rk}},\ (-1)^r\frac{2(n+1)}{m_{rk}}\right) \]

with the sheet \(H_{nk}^r\) attached to it is called an element of the surface \(F\in\mathfrak{A}\) and is denoted by \(G_{nk}^r(\Delta_{rk},m_{rk})\).

Definition. A surface \(F\in\mathfrak{A}\) is called an equivalent domain \(\Phi(\varphi,\rho\leqslant\infty)\) if, when a cut is made along the real axis of the sheet \(H\), dividing \(F\) into two simply connected parts \(F_1\) and \(F_2\), one can define a pair of quasiconformal mappings \(g_1(w)\) and \(g_2(w)\), with characteristics preserving the type of \(F\), such that \(g_1(F_1)=B_1,\ g_2(F_2)=B_2,\)

\(g_1(\gamma)=g_2(\gamma)=\pm\rho\), where \(\gamma\) is a boundary element \(F\), and for all \(x\), \(\varphi(x)=g_2[g_1^{-1}(x)]\).

2. We further assume that the quantity \(\frac12\Delta_{rk}-m_{rk}^{-1}\) is of order not less than \(m_{rk}^{-1}\) as \(k\to\pm\infty\). Let \(\beta_{nk}^{r}\) be the center of the rectangle \(R_{nk}^{r}\), and let \(w^*=L_{nk}^{r}(w)\), \(w^*=u^*+iv^*\), be the translation transformation, \(L_{nk}^{r}(\beta_{nk}^{r})=0\). Put
\[ \hat G_{nk}^{r}=L_{nk}^{r}\bigl(G_{nk}^{r}(\Delta_{rk},m_{rk})\bigr). \]

Lemma. One can construct a \(q\)-quasiconformal mapping \(\omega=g_{rk}(w^*)\) of the element \(\hat G_{nk}^{r}\) onto a univalent rectangle
\[ |\operatorname{Im}\omega|<m_{rk}^{-1},\qquad |\operatorname{Re}\omega|<(\pi m_{rk})^{-1}\ln m_{rk}\Delta_{rk}(1+\varepsilon_{rk}), \]
where \(\varepsilon_{rk}\to0\) as \(\Delta_{rk}m_{rk}\to\infty\), such that \(|g'_{rk}(w^*)|\equiv1\) on the lateral sides \(\hat R_{nk}^{r}\), and on the horizontal bases of \(\hat R_{nk}^{r}\), \(\hat R_{nk}^{r}=L_{nk}^{r}(R_{nk}^{r})\),
\[ g_{rk}\left(u^*\pm i\frac1{m_{rk}}\right) = \frac1{\pi m_{rk}}\ln \frac{ l_{rk}+u^*\left[1-\frac{2}{m_{rk}\Delta_{rk}}(1+2\lambda_{rk})\right] }{ l_{rk}-u^*\left[1-\frac{2}{m_{rk}\Delta_{rk}}(1+2\lambda_{rk})\right] } \pm i\frac1{m_{rk}}. \]

Here
\[ l_{rk}=\frac12\Delta_{rk}-\frac1{m_{rk}}\left(\frac32\lambda_{rk}+1\right), \qquad \lambda_{rk}<K=\mathrm{const}. \]

Theorem 1. Let \(\varphi(x)\), \(\varphi'(x)>0\), be a continuously differentiable gluing function of the half-strips \(B_1\) and \(B_2\), admitting conformal gluing. Then in the class \(\mathfrak A\) there exists a surface equivalent to the domain \(\Phi(\varphi,\rho\leq\infty)\).

Proof. On the interval \((-\rho,\rho)\) consider a continuous piecewise-linear function \(\bar x=\psi(x)\), \(\psi'(x)>0\), \(\psi(\pm\rho)=\pm\rho\), which for all \(x\) satisfies the condition
\[ \max\left(\frac{\varphi'(x)}{\psi'(x)},\frac{\psi'(x)}{\varphi'(x)}\right)<K, \tag{1} \]
where \(K=\mathrm{const}\). Let \(\{x_k\}\), \(x_k<x_{k+1}\), \(k=0,\pm1,\pm2,\ldots\), \(\lim_{k\to\pm\infty}x_k=\pm\rho\), be the set of discontinuity points of \(\psi'(x)\). The choice of such a function under the hypotheses of the theorem is possible. The deformation \(\xi=\eta(z)=\varphi[\psi^{-1}(x)]+iy\) for \(\operatorname{Im}z>0\) and \(\eta(z)\equiv z\) for \(\operatorname{Im}z<0\), under condition (1), provides a \(q\)-quasiconformal transition from \(\Phi(\psi,\rho\leq\infty)\) to \(\Phi(\varphi,\rho\leq\infty)\). Let \(\{\alpha_k\}\), \(\alpha_k<\alpha_{k+1}\), \(k=0,\pm1,\pm2,\ldots\), \(\lim_{k\to\pm\infty}\alpha_k=\pm\infty\), be a sequence of real numbers, whose choice we shall specify below. For each \(k\) define a collection of elements
\[ \{G_{nj_{rk}}^{r}=G_{nj_{rk}}^{r}(\Delta_{rj_{rk}},m_{rj_{rk}})\},\qquad j_{rk}=n_{rk},\,n_{rk}+1,\ldots,n_{rk+1}-1, \]
\[ \Delta_{rj_{rk}}=\Delta_{rj_{rk+1}},\qquad m_{rj_{rk}}=m_{rj_{rk+1}} \quad\text{for}\quad n_{rk}\leq j_{rk}\leq n_{rk+1}-2, \]
connected by translation transformations
\[ G_{ni'}^{r}=T_{rk}^{(i')}(G_{nn_{rk}}^{r}),\qquad G_{i''j_{rk}}^{r}=\hat T_{rk}^{(i'')}(G_{1j_{rk}}^{r}), \]
where \(T_{rk}^{(i')}\) and \(\hat T_{rk}^{(i'')}\) are superpositions of \(i'\) and \(i''\) transformations, respectively,
\[ T_{rk}=w+\Delta_{r n_{rk}},\qquad \hat T_{rk}=w+(-1)^r\frac{2i}{m_{r n_{rk}}}. \]
The corresponding rectangles \(\{R_{nj_{rk}}^{r}\}\) fill the half-strips
\[ S_{rk}:\quad (-1)^r v>0,\qquad \alpha_k<u<\alpha_{k+1}, \]
univalently, and \(R_{1n_{rk}}^{r}\) is the rectangle
\[ v\in\left(0,\,(-1)^r\frac{2i}{m_{r n_{rk}}}\right),\qquad u\in(\alpha_k,\alpha_k+\Delta_{r n_{rk}}). \]

Let
\[ F=\bigcup_{k,r}F_{rk},\qquad F_{rk}=\bigcup_{n,j_{rk}}\bar G_{nj_{rk}}^{r}. \]
Obviously, \(F\in\mathfrak A\). Define on the interval \((0,\infty)\) a continuous function \(h(\xi)>0\), subjecting it to the conditions
\[ h(\xi)\to\infty,\qquad \frac1{\xi}h(\xi)\to0 \quad\text{as}\quad \xi\to\infty. \]
Put
\[ \Delta_{rj_{rk}}=\frac1{h(m_{rk})}, \]
where \(m_{rk}=m_{r n_{rk}}\), for all \(k\) and \(r\). In this case each element belonging to \(F_{rk}\) is transformed \(q\)-quasiconformally onto a univalent ...

rectangle with base length

\[ d_{rk}=\frac{2}{\pi m_{rk}}\ln \frac{m_{rk}}{h(m_{rk})}\,(1+\varepsilon_{rk}) \]

and height \(2m_{rk}^{-1}\). If \(m_{rk}\) is now determined from the equalities

\[ \frac{2}{\pi}\frac{h(m_{rk})}{m_{rk}}\ln \frac{m_{rk}}{h(m_{rk})}\,(1+\varepsilon_{rk}) =\frac{\delta_{rk+1}-\delta_{rk}}{\alpha_{k+1}-\alpha_k}, \tag{2} \]

where \(\delta_{1k}=x_k\) and \(\delta_{2k}=\psi(x_k)\), then the surface \(F_{rk}\) is mapped \(q\)-quasiconformally, under a suitable normalization of the mappings of the lemma, onto half-strips in the plane \(z_1\),

\[ \hat S_{rk}:\quad (-1)^r\operatorname{Im} z_1>0,\qquad \delta_{rk}<\operatorname{Re}z_1<\delta_{rk+1}. \]

Denote these mappings by \(z_1=v_{rk}(w)\) and note that \(v'_{rk}(w)=g'_{rk}(w)\), where \(g_{rk}(w)\) is the mapping of the lemma. In \(\hat S_{rk}\) consider an additional deformation \(z=v^*_{rk}(z_1)\), transforming \(\hat S_{rk}\) into itself, which outside the semicircles

\[ P_{rj'_rk}:\quad |z_1-\delta_{rk}-d_{rk}(j'_{rk}+1/2)|<\tfrac12 d_{rk},\qquad (-1)^r\operatorname{Im}z_1>0, \]

\(j'_{rk}=j_{rk}-n_{rk}\), coincides with the identity mapping, and inside the latter

\[ v^*_{rk}(z_1)=\mu^{-1}\{\chi[\mu(z_1)]\}. \]

Here (we omit the indices) \(z_2=\mu(z_1)\) is a \(q\)-quasiconformal mapping of the semicircle \(P_{rj'_rk}\) onto the half-plane \((-1)^r\operatorname{Im}z_2>0\),

\[ \mu(\delta_{rk}+d_{rk}(j'_{rk}+1))=1,\qquad \mu(\delta_{rk}+d_{rk}j'_{rk})=0, \]

\[ \mu(\delta_{rk}+d_{rk}(j'_{rk}+1/2)+(-1)^r i d_{rk}/2)=(-1)^r i;\qquad |\mu'(\operatorname{Re}z_1)|=1/d_{rk}, \]

and \(\chi(z_2)\) is a quasiconformal mapping of the form

\[ \chi(z_2)=L[\hat v_{rk}^{-1}(\operatorname{Re}z_2)]+i\operatorname{Im}z_2, \]

where \(\hat v_{rk}(w)=\mu[v_{rk}(w)]\), and \(L(w)\) is an integral linear function,

\[ L[\hat v_{rk}^{-1}((0,1))]=(0,1),\qquad L'(w)=1/\Delta_{rj_rk}, \]

for points of the half-strip

\[ 0<\operatorname{Re}z_2<1,\qquad (-1)^r\operatorname{Im}z_2>0, \]

and \(\chi(z_2)\equiv z_2\) for the remaining points of the half-plane \((-1)^r\operatorname{Im}z_2>0\). The resulting mapping \(z=f_{rk}(w)\) maps \(F_{rk}\) quasiconformally onto the half-strips \(\hat S_{rk}\), with constant boundary stretching, and at the points of the boundary half-lines of \(F_{rk}\)

\[ |f'_{rk}(w)|\equiv 1. \]

Thus the function \(z=f_r(w)\), \(f_r(w)=f_{rk}(w)\) for \(w\in F_{rk}\), maps the selected surface quasiconformally onto \(\Phi(\psi,\rho\leq\infty)\). Since \(v'_{rk}=g'_{rk}\), we have

\[ \frac{1}{K}\frac{1}{m_{rk}\Delta_{rj_rk}}<v'_{rk}<K, \]

where \(K\) is a constant independent of \(k\) and \(r\), for all \(u\in(\alpha_k,\alpha_{k+1})\). Therefore, for \(z_2\in \mu\left(\bigcup_{j'_rk}P_{rj'_rk}\right)\),

\[ p_\chi(z_2)<\max\left(\frac{L'}{\mu'v'_{rk}},\frac{\mu'v'_{rk}}{L'}\right) <K\frac{m_{rk}}{h(m_{rk})\ln\bigl(m_{rk}/h(m_{rk})\bigr)}. \tag{3} \]

By construction, \(p_{f_r}(z)\) at the points of the set \(\bigcup_{k,j'_rk}P_{rj'_rk}\) does not exceed the right-hand side of (3), possibly with another constant factor, and outside it is uniformly bounded. Under the conditions of the theorem there exists a pair of functions regular and univalent in the half-strips \((-1)^r\operatorname{Im}\zeta\geq 0\), \(|\operatorname{Re}\zeta|<\rho\leq\infty\),

\[ \tau=t_r(\zeta), \]

which map \(\Phi(\varphi,\rho\leq\infty)\) onto the disk \(|\tau|<R\leq\infty\). Put

\[ t(\zeta)=t_r(\zeta)\quad \text{for } \zeta\in B_r, \qquad \zeta=\eta(w)=\eta[f_r(w)]. \]

Let \(\tau_1=t_1(\tau)\) be a quasiconformal mapping of the annulus \(R_0<|\tau|<R\) onto the annulus \(R_0<|\tau_1|<R_1\), with characteristic

\[ p=p_{\hat\eta}(\tau)\quad \text{for } \tau\in t\left[\eta\left(\bigcup_{k,j'_rk}P_{rj'_rk}\right)\right]\cap(R_0<|\tau|<R)=E_r, \]

and \(p=1\) on the complementary set. By the Belinskii–Helly theorem \((^{2,3})\),

\[ \left|\ln\frac{R}{R_1}\right| \leq \iint_{R_0<|\tau|<R}\frac{p-1}{|\tau|^2}\,d\sigma_\tau < \sum_{r=1,2}\iint_{E_r}\frac{p_{\hat\eta}(\tau)}{|\tau|^2}\,d\sigma_\tau < K\sum_{r=1,2}\iint_{E_r}\frac{p_{f_r}(\tau)}{|\tau|^2}\,d\sigma_\tau . \tag{4} \]

Here \(d\sigma_\tau\) is the area element in the \(\tau\)-plane. To estimate the integral on the right-

of the right-hand part of (4), we pass to the \(\zeta\)-plane. We have

\[ \left|\ln {R\over R_1}\right| < K \sum_{r=1,2}\iint_{t^{-1}(E_r)} p_{fr}(\zeta)\left|{d\ln t(\zeta)\over d\zeta}\right|^2\,d\sigma_\zeta < K \sum_{r,k\ge k_0} \operatorname{mes}\eta\!\left(\bigcup_{j'_{rk}} P_{rj'_{rk}}\right)p_{rk}M_{rk}, \tag{5} \]

where \(p_{rk}=\max p_{fr}(\zeta)\), \(M_{rk}=\max\left|d\ln t(\zeta)/d\zeta\right|^2\) for
\(\zeta\in\eta\left(\bigcup_{j'_{rk}}P_{rj'_{rk}}\right)\), and the summation is carried out over those sets
\(\eta\left(\bigcup_{j'_{rk}}P_{rj'_{rk}}\right)\) for which
\(\eta\left(\bigcup_{j'_{rk}}P_{rj'_{rk}}\right)\cap t^{-1}(E_r)\ne0\). Since, by the definition of the mapping \(\zeta=\eta(z)\), we have

\[ \operatorname{mes}\eta\!\left(\bigcup_{j'_{kr}} P_{rj'_{rk}}\right) \le \max_{x\in\delta_{rk}}\left|{\partial\eta\over\partial x}\right| \operatorname{mes}\bigcup_{j'_{rk}}P_{rj'_{rk}} < K{\delta_{rk+1}-\delta_{rk}\over m_{rk}} \ln {m_{rk}\over h(m_{rk})}, \tag{6} \]

then, assuming \(\delta_{rk+1}-\delta_{rk}<O(1)\), from (3), (5), and (6) we finally obtain

\[ \left|\ln {R\over R_1}\right| < K\sum_{r,k\ge k_0}{M_{rk}\over h(m_{rk})}. \tag{7} \]

After \(\psi(x)\) has been chosen, the multiplier \(\delta_{rk+1}-\delta_{rk}\) in the right-hand side of (2) is fixed, and moreover
\[ {h(\xi)\over \xi}\ln{\xi\over h(\xi)}\to0 \quad\text{as}\quad \xi\to\infty. \]
Choose \(\alpha_{k+1}-\alpha_k\) so that, for \(m_{rk}\), determined from (2), the inequality
\[ h(m_{rk})>\max_{r=1,2} M_{rk}\lambda_k \]
holds, where
\[ \sum_k {1\over \lambda_k}<\infty. \]
Then it follows from (7) that, for these \(\{\alpha_k\}\) and the chosen \(\psi(x)\) and \(h(\xi)\), \(R\) and \(R_1\) are simultaneously finite or infinite. We now define a quasiconformal mapping of the annulus
\[ R_0<|\tau_1|<R_1 \]
onto the annulus
\[ R_0<|\tau_2|<R_2 \]
with characteristic \(\hat p=p_{fr}(\tau_1)\) outside the set \(t_1(E_r)\), and \(\hat p=1\) on the complement. This mapping is \(q\)-quasiconformal; therefore
\[ {1\over K}R_1<R_2<KR_1, \]
where \(K=\mathrm{const}\). Thus the resulting quasiconformal mapping of the annulus \(R_0<|\tau|<R\) with characteristic \(p_{fr}(\tau)\) changes its modulus by no more than \(K=\mathrm{const}\) times. The theorem is proved.

1. Denote by \(\mathfrak A[\Phi(\varphi,\rho<\infty)]\) the class of all surfaces from \(\mathfrak A\) equivalent to \(\Phi(\varphi,\rho<\infty)\). By what has been proved, it is nonempty and contains, for example, surfaces equivalent to \(\Phi(\varphi,\rho<\infty)\) obtained for different choices of the approximating function \(\psi(x)\), the function \(h(\xi)\), and the sequence \(\{\alpha_k\}\). The validity of the following theorem is obvious.

Theorem 2. All surfaces of the class \(\mathfrak A[\Phi(\varphi,\rho<\infty)]\) are of hyperbolic type.

Various sufficient type criteria for \(\Phi(\varphi,\rho=\infty)\), each time applicable to all surfaces from \(\mathfrak A[\Phi(\varphi,\rho=\infty)]\), were obtained in \((4\text{–}9)\).

In conclusion I express my gratitude to A. A. Goldberg for a number of valuable comments.

Received
24 V 1961

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