Abstract
Full Text
MATHEMATICS
D. B. Potyagailo
ON THE TYPE OF GLUING OF A STRIP
(Presented by Academician M. A. Lavrent’ev, 1 II 1961)
1. In the present note criteria are indicated for the parabolic type of gluing of a strip, giving a strengthening of the results obtained in \((^{1,2})\). Let
\(\bar{x}=g(x)\) be a continuously differentiable topological correspondence between the points of the boundary lines of the strip \(S: 0<y<1,\ -\infty<x<\infty\) in the plane \(z=x+iy\), admitting a quite definite conformal gluing \((^3)\). The function \(g(x)\), as is customary, will be called the gluing function, and by \(S\) with the indicated correspondence on the boundary we shall understand a certain Riemann surface \(S(g)\). Suppose that the function \(w=T(z)\) effects a conformal sealing-up of \(S(g)\), transforming the latter into a domain filling a disk \(|w|<R\leq\infty\) one-sheetedly, in such a way that \(T(x)=T[g(x)+i]\). Then \(S(g)\) is called a surface of parabolic or hyperbolic type according as \(R=\infty\) or \(R<\infty\). Let us note that if \(g_1(x)\) and \(g_2(x)\) are two gluing functions of the strip satisfying, for all \(x\), the conditions
\[ \max\left(\frac{g_1'(x)}{g_2'(x)},\ \frac{g_2'(y)}{g_1'(x)}\right)<K,\qquad |g_1(x)-g_2(x)|<K, \tag{1} \]
where \(K\) is a constant, then one can construct a \(q\)-quasiconformal mapping which reduces the problem of gluing \(S\) with gluing function \(g_1(x)\) to the problem of gluing \(S\) with gluing function \(g_2(x)\). In this case the type of \(S(g_1)\) and \(S(g_2)\) is one and the same.
2. Let \(\varphi(x)\), \(\varphi'(x)>0\), be a continuous piecewise-linear function, defined for all \(x\in(-\infty,\infty)\) and such that: a) the set \(M\) of abscissae of the break points of \(\varphi(x)\) has the only limit point at \(\infty\), and, for \(x'\notin M\), \(\varphi(x')=g(x')\); b) for all \(x\) the conditions (1) are fulfilled for \(\varphi(x)\) and \(g(x)\). The final choice of \(\varphi(x)\) will be specified below. Denote \(\{x_k\}=M\cap[0,\infty)\), \(x_0=0\), \(x_k<x_{k+1}\), \(k=0,1,2,\ldots\), and let \(\gamma_k\) be the rectilinear segments in \(S\) joining the points \(x_k\) and \(\varphi(x_k)+i=\varphi_k+i\). Let \(\{B_k\}\) be quadrilaterals, each of which is bounded by a pair of segments \(\gamma_k,\gamma_{k+1}\) and by segments of the boundary lines of \(S\) lying between them. Put \(n_k=x_{k+1}-x_k\), \(m_k=\varphi_{k+1}-\varphi_k\), and \(S_0=\bigcup_{k=0}^{\infty}B_k\). The modulus \(\tilde{\mu}\) of the doubly connected domain \(S_0(\varphi)\), conformally equivalent to \(S_0\) with gluing function \(\varphi(x)\), is obviously connected with the modulus of the part \(S_0(g)\) of the surface \(S(g)\), obtained from it by removing a certain disk, by the inequality
\[ \frac{1}{K}\,\mu<\mu[S_0(g)]<K\tilde{\mu}. \]
This follows from the remark in § 1 and the choice of \(\varphi(x)\). Let \(\{B_k\}=\mathfrak{M}_1\cup\mathfrak{M}_2\), where
\(\mathfrak{M}_1=\mathfrak{M}[B_k,\ \max(m_k/n_k,\ n_k/m_k)\geq 1+\delta]\) and
\(\mathfrak{M}_2=\mathfrak{M}[B_k,\ \max(m_k/n_k,\ n_k/m_k)<1+\delta]\), \(\delta>0\). For each \(B_k\in\mathfrak{M}_1\) (for simplicity we omit the indices) with vertices \(a,b,c\), and \(d\) (see Fig. 1, II) we indicate a quasiconformal gluing which transforms it into a ring. Extend the lateral sides of \(B_k\) until they meet at the point \(o\). Assuming that the angle at the vertex \(o\) is small, we shall distinguish two cases: 1) \(|\pi/2-\theta|<\theta_0\) and 2) \(|\pi/2-\theta|\geq\theta_0>0\), where \(\theta\) is the angle at the vertex \(c\). In case 2), assuming for definiteness that \(\overline{bc}>\overline{ad}\) and \(\theta\leq\pi/2-\theta_0\), we fix on the side \(dc\) points \(p_1\) and \(p_2\) such that \(\overline{bp_1}=\overline{bc}\) and \(\overline{ob}=\overline{op_2}\).
One can construct a quasiconformal mapping \(\tau=\mu_{k'}(z)\), taking the triangle \(p_1bc\) onto the triangle \(p_1bp_2\), which leaves the side \(p_1b\) fixed, \(\mu_{k'}(c)=p_2\), and has constant stretching at the points of the side \(bc\). Since in the indicated triangles the angles at the vertex \(b\) satisfy the condition \(\min(\theta',\theta'')>\theta_0\), the characteristic of this mapping is \(p_1(z)<K_1\), where \(K_1\) depends only on \(\theta\), and \(K_1(\theta)\to\infty\) as \(\theta\to0\). Let now \(q_1\) and \(q_2\) be points on the side \(ab\) such that \(\overline{ad}=\overline{dq_1}\) and \(\overline{od}=\overline{oq_2}\), and let \(\tau=\chi_{k'}(z)\) be an analogous quasiconformal mapping taking the triangle \(adq_1\) onto the triangle \(q_2dq_1\). Here the side \(dq_1\) remains fixed, \(\chi_{k'}(a)=q_2\), and \(|\chi_{k'}(z)|=\mathrm{const}\) at the points of the side \(ad\). By the same considerations its characteristic \(p_2(z)<K_2\), where \(K_2(\theta)\to\infty\) as \(\theta\to0\). The function \(\tau=f_{k'}(z)\), \(f_{k'}(z)=\mu_{k'}(z)\) for \(z\in\Delta_{k'}=(p_1bc)\), \(f_{k'}(z)=\chi_{k'}(z)\) for \(z\in\Delta_{k'}=(adq_1)\), and \(f_{k'}(z)\equiv z\) for the remaining points \(B_{k'}\), realizes a quasiconformal mapping of \(B_{k'}\) onto the quadrilateral \(\hat B_{k'}=(q_2dp_2b)\) with piecewise-continuous characteristic \(p(z)\), \(p(z)<\max(K_1,K_2)\). Let \(\sigma_1\) and \(\sigma_2\) be smooth Jordan arcs in \(\hat B_{k'}\) with endpoints at the points \(q_2,d\) and \(p_2,b\), respectively, dividing \(\hat B_{k'}\) into the regions \(\mathfrak B'_{k'}\), \(\mathfrak B''_{k'}\), and \(\hat B_{k'}\setminus(\mathfrak B'_{k'}\cup\mathfrak B''_{k'})\). Choosing \(\sigma_1\) and \(\sigma_2\) so that the angles which they form with the segments \(q_2d\) and \(p_2b\) are, for all \(k'\), no smaller than a certain positive constant, and moreover \(\bar\sigma_1=O(\overline{q_2d})\), \(\bar\sigma_2=O(\overline{p_2b})\), it is easy to construct in \(\hat B_{k'}\) an additional \(q\)-quasiconformal deformation \(\tau^*=t(\tau)\), which in \(\hat B_{k'}\setminus(\mathfrak B'_{k'}\cup\mathfrak B''_{k'})\) coincides with the identity transformation, while inside \(\mathfrak B'_{k'}\) and \(\mathfrak B''_{k'}\) it takes, with constant stretching, the boundary segments \(q_2d\) and \(p_2b\) to arcs of circles of radii \(\overline{od}\) and \(\overline{op_2}\), respectively. By means of the superposition \(w=h(\tau^*)\)—a conformal compression with coefficient \(1/\overline{ob}\), a logarithm, and an integral linear function—we pass to a vertical rectangle \(G_{k'}\) in the plane \(w\), \(w=u+iv\), of unit height and base \((u_{k'},u_{k'+1})\). The transformation \(w^*=\exp(2\pi w)\) turns \(G_{k'}\) into an annulus. The indicated gluing is preserved for \(\theta\le\pi/2-\theta_0\), \(\overline{bc}<\overline{ad}\), and also for \(\theta\ge\pi/2+\theta_0\), \(\overline{bc}\le\overline{ad}\). In all cases the length of the base of the rectangle \(G_{k'}\), as an elementary calculation shows, satisfies the inequality
\[ d_{k'}>K\,\frac{[\max(m_{k'}/n_{k'},\,n_{k'}/m_{k'})-1]\min(n_{k'},m_{k'})}{\min(\gamma_{k'}^{2},\gamma_{k'+1}^{2})\ln\max(m_{k'}/n_{k'},\,n_{k'}/m_{k'})}, \quad \text{where } K \text{ is a constant.} \tag{2} \]
Consider now case 1). The preliminary quasiconformal transformation \(f_{k'}(z)\) is defined here differently. Assuming that \(\pi/2-\theta_0\le\theta\le\pi/2\), \(\overline{ad}<\overline{dc}\) and \(\overline{bc}>\overline{ad}\), mark on the side \(ab\) the point \(p_1\), \(\overline{p_1b}=\overline{bc}\), and map quasiconformally, by means of the function \(\tau=\tilde\mu_{k'}(z)\), the triangle \(p_1bc\) onto the triangle \(p_1p_2c\) (Fig. 1, III), where \(p_2\) is a point lying on the continuation of the side \(ab\), \(\overline{op_2}=\overline{oc}\). We choose this mapping so that the
![Fig. 1]
Fig. 1
the side \(p_1c\) remained fixed, \(\widetilde{\mu}_{k'}(b)=p_2\), and \(|\widetilde{\mu}_{k'}(z)|=\operatorname{const}\) at the points of the side \(bc\). Since the angles of these triangles are uniformly bounded below by a positive constant, \(\widetilde{\mu}_{k'}(z)\) is a \(q\)-quasiconformal mapping. Next, in the triangle \(adq_1\) we define a \(q\)-quasiconformal mapping \(\widetilde{\chi}_{k'}(z)\) carrying it onto the triangle \(aq_2q_1\), where \(q_1\) and \(q_2\) are points lying respectively on \(dc\) and \(od\), \(\overline{ao}=\overline{oq_2}\) and \(\overline{ad}=\overline{dq_1}\). In this case the side \(aq_1\) remains fixed, \(\widetilde{\chi}_{k'}(d)=q_2\), and \(|\widetilde{\chi}_{k'}(z)|=\operatorname{const}\) at the points of the side \(ad\). We now put \(f_{k'}(z)=\widetilde{\mu}_{k'}(z)\) for \(z\in \Delta_{k'}=(p_1bc)\), \(f_{k'}(z)=\widetilde{\chi}_{k'}(z)\) for \(z\in \Delta'_{k'}=(adq_1)\), and \(f_{k'}(z)\equiv z\) for the remaining points. The further transformations are the same as in case 2). For all possible forms here of the original quadrilateral \(B_{k'}\), the estimate for the length of the base \(d_{k'}\) is the same as in (2).
Let \(G_{k''}\) be a rectangle, conformally equivalent to \(B_{k''}\in \mathfrak{M}_2\), of unit height, with base \((u_{k''},u^0_{k''+1})\) and with the identical correspondence between the points of the horizontal bases. We may assume that all \(\{G_{k'},G_{k''}\}=\{G_k\}\) fill, in a one-to-one manner, the rectangle \(R\), \(0<u<\rho\leq\infty\), \(0<v<1\), and that between the points of the adjacent sides of each pair of rectangles \(G_k\) and \(G_{k+1}\), \(k=0,1,2,\ldots\), a homeomorphic correspondence is established. Assigning to each point \(u_k\) the interval \((u_k,u_k^0)\), \(u_k^0\in(u_k,u_{k+1})\), where \((u_k,u_{k+1})\) is the base of \(G_k\), we define a quasiconformal gluing of \(\{G_k\}\), identical outside the rectangles \(\widehat{G}_k=\{u\in(u_k,u_k^0),\,v\in(0,1)\}\), and inside quasiconformal, preserving the abscissae of the points. Put
\[ I_n=\bigcup_{k=1}^{n}(u_k,u_k^0). \]
Let \(w=H(z)\) be the resulting homeomorphic mapping of \(S_0\) onto \(R\), quasiconformal with characteristic \(p(w)\). If \(R_n=\{u\in(0,u_{n+1}),\,v\in(0,1)\}\), then, by Grötzsch’s principle,
\[ \mu(R_n)>\sum_{k=1}^{n}\mu(G_k\setminus \widehat{G}_k) \]
and, consequently, (4),
\[ \mu(R_n)> \sum_{(k')}\int_{u_{k'}^0}^{u_{k'+1}} \frac{du}{\displaystyle \int_{\Gamma_u} p\,\frac{du}{dn}\,ds} + \sum_{(k'')}\int_{x_{k''}'}^{x_{k''}''} \frac{dx}{\displaystyle \int_{\Gamma_x} \frac{dx}{dn'}\,ds'}. \tag{3} \]
Here \(\Gamma_u\) are vertical segments in \(G_{k'}\subset R_n\), and \(dn\) and \(ds\) are the elements of the normal and of arclength on \(\Gamma_u\). The section \(\Gamma_x\) coincides with the rectilinear segment in \(B_{k''}\) joining the points \(x\) and \(\varphi(x)+i\); \(dn'\) and \(ds'\) are the elements of the normal and of arclength on \(\Gamma_x\), and \((x_{k''}',x_{k''}'')\) is the base of the quadrilateral formed by all segments \(\Gamma_x\subset B_{k''}\subset H^{-1}(R_n)\) for which \(\Gamma_x\cap H^{-1}(G_{k''})=0\). On such a segment (Fig. 1, \(I\)) we have
\[ dx/dn'\leq \sqrt{1+[\varphi(x)-x]^2}\,\{1+[\varphi'(x)]^{-1}\}, \]
therefore
\[ \sum_{(k'')}\int_{x_{k''}'}^{x_{k''}''} \frac{dx}{\displaystyle \int_{\Gamma_x}\frac{dx}{dn'}\,ds'} \geq \frac12 \sum_{(k'')}\int_{x_{k''}'}^{x_{k''}''} \frac{\min(dx,d\varphi)}{1+[\varphi(x)-x]^2}. \tag{4} \]
Since on \(\Gamma_u\), \(du/dn=1\), putting \(\Gamma'_u=\Gamma_u\cap H(\Delta'_{k'}\cup\Delta_{k'}\cup \mathfrak{B}'_{k'}\cup\mathfrak{B}''_{k'})\), we have, on \(\Gamma_u\subset G_{k'}\setminus \widehat{G}_{k'}\),
\[ \int_{\Gamma_u}p(w)\frac{du}{dn}\,ds = \int_{\Gamma'_u}p(w)\frac{du}{dn}\,ds + \int_{\Gamma_u\setminus\Gamma'_u}\frac{du}{dn}\,ds < 1+\int_{\Gamma'_u}p(w)\,ds. \tag{5} \]
Since \(|dh/d\tau^*|<1\), it follows that
\[ \int_{\Gamma'_u}p(w)\,ds < \max(K_1,K_2)\int_{\Gamma'_u}ds < K\max(K_1,K_2)\max(m_{k'},n_{k'}), \]
where \(K\) is a constant independent of the choice of the section \(\Gamma_u\). Consequently, the right-hand side in (5) is uniformly bounded for a suitable choice of the function \(\varphi(x)\). From (3), (4) there then follows the inequality
\[ \mu(R_n)>K'\sum_{(k')}\int_{u_{k'}^{0}}^{u_{k'}+1}du+\frac12\sum_{k''}\int_{x_{k''}^{\prime}}^{x_{k''}^{\prime\prime}} \frac{\min(dx,d\varphi)}{1+[\varphi(x)-x]^2}. \tag{6} \]
Let \(\varphi(x)\) be chosen so that the oscillation of \(1+[\varphi(x)-x]^2\) on each interval \((x_k,x_{k+1})\) does not exceed \(K=\mathrm{const}\), and, for all \(x\in(-\infty,\infty)\),
\[ \frac1K\{1+[\varphi(x)-x]^2\}<1+[g(x)-x]^2<K\{1+[\varphi(x)-x]^2\}. \tag{7} \]
Then from (6), (7), and (2), taking into account that
\[ \min(\bar\gamma_k^2,\bar\gamma_{k+1}^2)= \min\{1+[\varphi(x_k)-x_k]^2,\;1+[\varphi(x_{k+1})-x_{k+1}]^2\} \]
and that the stretching of the mapping \(H(z)\) is constant at the points of the horizontal bases \(B_{k'}\), we obtain
\[ \mu(R_n)>K''\left\{ \int_{E_n'}\frac{\max[g'(x),\,1/g'(x)]\,\min(dx,dg)} {[1+\psi^2(x)]\,\ln\max[g'(x),\,1/g'(x)]} + \int_{E_n''}\frac{\min(dx,dg)}{1+\psi^2(x)} \right\}+\varepsilon_n, \tag{8} \]
where \(\psi(x)=g(x)-x\), \(E_n'=E[x,\ \max(\varphi',\ 1/\varphi')\geq 1+\delta]\cap H^{-1}(\overline{R_n})\), \(E_n''=E[x,\ \max(\varphi',\ 1/\varphi')<1+\delta]\cap H^{-1}(\overline{R_n})\), and \(\varepsilon_n\) depends on the choice of \(I_n\), with \(\lim\varepsilon_n=0\) as \(\operatorname{mes} I_n\to0\). The sum of the integrals on the right-hand side of (8), up to a term uniformly bounded as \(n\to\infty\) and depending only on the choice of \(\varphi(x)\), coincides with the sum of integrals of the same kind taken over the sets \(\mathscr E_n'= \mathscr E[x,\ \max(g',\ 1/g')\geq 1+\delta]\cap H^{-1}(\overline{R_n})\) and \(\mathscr E_n''=\mathscr E[x,\ \max(g',\ 1/g')<1+\delta]\cap H^{-1}(\overline{R_n})\). Passing to the limit as \(n\to\infty\) and observing that \(\lim\mu(R_n)=\mu(R)\), we arrive at the following assertion.
Theorem. For the parabolic type of \(S(g)\), it is sufficient that at least one of the integrals diverge
\[ \int_{\mathscr E'}\frac{\max(dx,dg)} {[1+\psi^2(x)]\,\ln\nu(x)},\qquad \int_{\mathscr E''}\frac{\min(dx,dg)}{1+\psi^2(x)}, \]
where \(\psi(x)=g(x)-x\), \(\nu(x)=\max(g',\,1/g')\), and \(\mathscr E'\) and \(\mathscr E''\) are the sets on which, respectively, \(\nu(x)\geq 1+\delta\) and \(\nu(x)<1+\delta\), \(\delta>0\).
As a consequence of this theorem we obtain a result of L. I. Volkovyskii.
Corollary 1. For the parabolic type \(S(g)\), it is sufficient that the integral
\[ \int_{0}^{\infty}\frac{\min(dx,dg)}{1+\psi^2(x)} \]
diverge.
Corollary 2. Let \(g(x)\) be such that \(\psi(x)\leq O(1)\). Then, for the parabolic type \(S(g)\), it is sufficient that at least one of the integrals diverge
\[ \int_{\mathscr E'}\frac{\max(dx,dg)}{\ln\nu(x)},\qquad \int_{\mathscr E''}\min(dx,dg). \tag{9} \]
Let us note that under the hypotheses of Corollary 2 there exist functions leading to the hyperbolic type \(S(g)\). At the same time it is not known whether, in this case, the simultaneous convergence of the integrals (9) is a sufficient condition for the hyperbolic type \(S(g)\).
Lviv State University
named after Ivan Franko
Received
25 I 1961
References
- L. N. Volkovyskii, Matem. sborn., 18 (62), 285 (1946).
- R. Nevanlinna, Ann. Acad. Sci. Fenn., Ser. AI, 122 (1952).
- L. I. Volkovyskii, Ukr. matem. zhurn., 1, 39 (1951).
- D. B. Potyagailo, Ukr. matem. zhurn., 5, 459 (1953).