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MATHEMATICS
V. G. MIKHAL’CHUK
AN EXISTENCE THEOREM FOR RATIONAL FUNCTIONS ON RIEMANN SURFACES
(Presented by Academician I. N. Vekua, January 2, 1961)
- In order that on a closed Riemann surface \(R\) of genus \(g\) there exist a rational analytic function with poles of the first order at prescribed points \(p_\mu\) \((\mu=1,2,\ldots,n)\), respectively with principal parts \(a_\mu/z\), it is necessary and sufficient that the equality
\[ \sum_{\mu=1}^{n} a_\mu \varphi(p_\mu)=0, \tag{1} \]
hold, where \(\varphi(p)\) is an arbitrary covariant of the first kind on \(R\).
This assertion may be derived from the following relation (see \((^1)\), p. 204):
\[ \int_J f(z)\,dw = -\sum_{\nu=1}^{g} A'_\nu B_\nu = 2\pi i \sum_{\mu=1}^{n} a_\mu \varphi(p_\mu), \tag{2} \]
where \(J\) is a system of canonical cuts \(K_1, K_2,\ldots,K_{2g}\) of the surface \(R\); \(dw\) is an Abelian differential of the first kind with periods \(A'_\nu, B'_\nu\) along \(K_{2\nu-1}\), respectively \(K_{2\nu}\); \(df(z)\) is a complexly normalized Abelian differential of the second kind with principal parts \(-a_\mu/z^2\) at the points \(p_\mu\) \((\mu=1,2,\ldots,n)\) and with periods \(A_\nu, B_\nu\) along \(K_{2\nu-1}\), respectively \(K_{2\nu}\), with \(A_\nu=0\) (the condition of complex normalization). Here \(z\) denotes a local parameter of the surface \(R\).
The aim of what follows is to generalize the indicated assertion to the case of quasianalytic functions.
- By a quasianalytic function \(f(z)\) we mean a continuously differentiable solution of an equation of the form
\[ w_{\bar z}=B(z)\overline{w_z},\qquad |B(z)|<1. \tag{3} \]
Let \(df(z)\) be a complexly normalized quasianalytic differential (it can be made such analogously to the case of an analytic differential), belonging to equation (3), with poles of the second order at the points \(p_\mu\) \((\mu=1,2,\ldots,n)\) and, respectively, with principal parts
\(\alpha_\mu dZ_{x_\mu}(z,z_\mu)+i\beta_\mu dZ^1_{x_\mu}(z,z_\mu)\), where \(\alpha_\mu,\beta_\mu\) are real coefficients (see \((^2)\), p. 30). Further, let \(dw=du+i\,dv\) be an arbitrary quasianalytic differential of the first kind, and let \(dw_k\) \((k=1,2,\ldots,2g)\) be a basis of differentials of the first kind belonging to the equation
\[ w'_{\bar z}=-\overline{B(z)}\,\overline{w'_z}. \tag{4} \]
Let us note that the basis of quasianalytic differentials is a \(2g\)-dimensional real space, i.e., it has \(2g\) linearly independent components with respect to real coefficients and is not reduced to a \(g\)-dimensional complex space, since a differential belonging to equation (4), multiplied by a complex number, will not belong to this equation.
Let us now consider the integral
\[ H=\int_J f(z)\,dw+\overline{f(z)}\,d\overline{w}. \]
Preserving the notation for the periods of the preceding paragraph for the differentials \(df(z)\) and \(dw\), we obtain
\[ H=-2\operatorname{Re}\sum_{\nu=1}^{g} A'_{\nu}B_{\nu}. \tag{5} \]
The integrand of the integral \(H\) is a complete differential; therefore its residues can be computed (see (2), p. 30)
\[ H=-\int_J w\,df+\overline{w}\,d\overline{f} = 4\pi\sum_{\mu=1}^{n} \left[ \alpha_{\mu}\frac{\partial v(p_{\mu})}{\partial x} + \beta_{\mu}\frac{\partial u(p_{\mu})}{\partial x} \right]. \tag{6} \]
Theorem 1. In order that on the closed Riemann surface \(R\) there exist a single-valued quasianalytic function of equation (3) with poles of the first order at the points \(p_{\mu}\) \((\mu=1,2,\ldots,n)\), respectively with principal parts \(\alpha_{\mu}Z_{x_{\mu}}+\mathrm{i}\beta_{\mu}Z^{1}_{x_{\mu}}\) \((\mu=1,2,\ldots,n)\), it is necessary and sufficient that the inequality
\[ \sum_{\mu=1}^{n} \left[ \alpha_{\mu}\frac{\partial v(p_{\mu})}{\partial x} + \beta_{\mu}\frac{\partial u(p_{\mu})}{\partial x} \right] =0, \tag{7} \]
hold, where
\[ dw= \frac{\partial(u+iv)}{\partial x}\,dx + \frac{\partial(u+iv)}{\partial y}\,dy \]
is an arbitrary quasianalytic differential of the first kind, belonging to equation (4) on the surface \(R\).
Proof. Necessity follows from comparing formulas (5) and (6).
Sufficiency. Let \(df(z)\) be a complex-normalized quasianalytic differential of the second kind belonging to equation (3), with poles at the points \(p_{\mu}\) \((\mu=1,2,\ldots,n)\) and with principal parts respectively equal to \(\alpha_{\mu}dZ_{x_{\mu}}+\mathrm{i}\beta_{\mu}dZ^{1}_{x_{\mu}}\). Suppose that condition (7) is satisfied. Then, from comparing formulas (5) and (6), we obtain
\[ \operatorname{Re}\sum_{\nu=1}^{g} A'_{\nu}B_{\nu}=0. \tag{8} \]
Condition (8) is true for any quasianalytic differential of the first kind belonging to equation (4); hence it holds also for the basis differentials \(dw_k\) \((k=1,2,\ldots,2g)\), belonging to the same equation, i.e.,
\[ \operatorname{Re}\sum_{\nu=1}^{g} B_{\nu}A_{k\nu}=0 \qquad (k=1,2,\ldots,2g), \]
where \(A_{kv}\) is the period of the differentials \(dw_k\) with respect to the paths \(K_{2v-1}\). If
\[ B_v=b_v^{(1)}+ib_v^{(2)}, \qquad A_{kv}=a_{kv}^{(1)}+ia_{kv}^{(2)}, \]
then the last system can be written in the form
\[ \sum_{v=1}^{g}\left[a_v^{(1)}b_{kv}^{(1)}-b_v^{(2)}a_{kv}^{(2)}\right]=0 \qquad (k=1,2,\ldots,2g). \tag{9} \]
The determinant of the system (9) can differ only in sign from the determinant of the system
\[ a_k^{(1)}=\sum_{v=1}^{2g} c_v a_{vk}^{(1)}, \]
\[ a_k^{(2)}=\sum_{v=1}^{2g} c_v a_{vk}^{(2)} \qquad (k=1,2,\ldots,g), \tag{10} \]
where \(c_v\) are real coefficients determining a certain differential
\[ dw=\sum_{v=1}^{2g} c_v dw_v \tag{11} \]
with periods \(a_k^{(1)}+ia_k^{(2)}\) with respect to the paths \(K_{2k-1}\). But the determinant of the system (10) is different from zero by virtue of the uniqueness of the determination of the differential \(dw\) by arbitrarily prescribed periods \(a_k^{(1)}+ia_k^{(2)}\). Consequently, the determinant of the system (9) is different from zero, and therefore the system has only the trivial solution \(B_v=0\). Hence the function \(f(z)\) is single-valued on \(R\).
- The assertion given in no. 1 concerning the existence on the surface \(R\) of a rational analytic function with poles of the first order can be generalized also to the case of poles of higher orders. If the function \(f(z)\) has at the points \(p_\mu\) \((\mu=1,2,\ldots,n)\) poles of order \(m_\mu\) with principal parts
\[ \frac{a_{m_\mu}^{(\mu)}}{z^{m_\mu}}+\frac{a_{m_\mu-1}^{(\mu)}}{z^{m_\mu-1}}+\cdots+\frac{a_1^{(\mu)}}{z}, \]
then instead of condition (1) we shall have
\[ \sum_{\mu=1}^{n}\left[ \frac{a_{m_\mu}^{(\mu)}}{(m_\mu-1)!}\, \frac{d^{m_\mu-1}}{dz^{m_\mu-1}}\varphi(p_\mu) +\cdots+ \frac{a_2^{(\mu)}}{1!}\,\frac{d}{dz}\varphi(p_\mu) +a_1^{(\mu)}\varphi(p_\mu) \right]=0. \tag{12} \]
This is not difficult to verify by computing the residues of the expression \(f(z)\,dw\). It should be noted that the quantities \(a_{m_\mu}^{(\mu)}, a_{m_\mu-1}^{(\mu)}, \ldots, a_1^{(\mu)}\) depend in a definite way on the choice of the local parameter; however, each bracketed expression in relation (12) is invariant with respect to the choice of the local parameter.
An analogous generalization also holds for quasianalytic functions, namely, the following theorem holds:
Theorem 2. In order that on a closed Riemann surface \(R\) there exist a rational quasianalytic function satisfying
…of (3) with poles at the points \(p_\mu\) \((\mu=1,2,\ldots,n)\) of order \(m_\mu\), and in them with principal parts
\[ \sum_{j=1}^{m_\mu}\left[\alpha_j^{(\mu)} Z_{x_\mu^j}(z,z_\mu)+i\beta_j^{(\mu)} Z_{y_\mu^j}(z,z_\mu)\right], \]
it is necessary and sufficient that the equality
\[ \sum_{\mu=1}^{n}\left\{\sum_{j=1}^{m_\mu}\left[ \frac{\alpha_j^{(\mu)}}{(j-1)!}\frac{\partial^{j-1}v(p_\mu)}{\partial x^{j-1}} + \frac{\beta_j^{(\mu)}}{(j-1)!}\frac{\partial^{j-1}u(p_\mu)}{\partial x^{j-1}} \right]\right\}=0, \tag{13} \]
where \(dw=du+i\,dv\) is an arbitrary quasianalytic differential of the first kind belonging to equation (4) on the surface \(R\).
For the proof of Theorem 2 one should require the existence, in a neighborhood of the points \(p_\mu\), of derivatives of the functions \(\dot B(z)\) (see equation (3)) of order \((m_\mu-1)\).
- Theorems 1 and 2 hold for solutions of a more general elliptic system of differential equations, namely
\[ v_y=au_x+bu_y, \]
\[ v_x=cu_x+du_y, \]
which in complex form may be written as follows:
\[ w_{\bar z}=A_1(z)w_z+B_1(z)\bar w_{\bar z}, \tag{14} \]
where
\[ A_1=-\frac{p_1(p^2-1)}{(pp_1+1)(p+p_1)}e^{2i\theta}, \qquad B_1=\frac{p(p_1^2-1)}{(pp_1+1)(p+p_1)}e^{2i\theta_1}, \]
\(p,\theta;\ p_1,\theta_1\) are the characteristics of the quasiconformal mapping \(w=F(z)\) (see (2)). This follows from the following considerations. If on the Riemann surface \(R\) a solution of differential equation (14) is given, then by choosing a new local parameter \(\zeta=\varphi(z)\), which in each parametric circle \(|z|<1\) satisfies the equation
\[ \varphi_{\bar z}=A(z)\varphi_z,\qquad A=-\frac{p-1}{p+1}e^{2i\theta}, \]
one passes to the function \(w=f(\zeta)\), which is a solution of equation (3) with respect to the local parameter \(\zeta\) on the surface \(R_1\), obtained from \(R\) by changing the local parameters.
Perm State Pedagogical Institute
Received
19 XII 1960
REFERENCES
- R. Nevanlinna, Uniformization, IL, 1955.
- V. G. Mikhal’chuk, Scientific Notes of Perm University, 16, no. 3, 27 (1958).