Reports of the Academy of Sciences of the USSR
1968. Volume 183, No. 4

UDC 517.543

MATHEMATICS

S. L. KRUSHKAL

ON THE PROBLEM OF MODULI OF RIEMANN SURFACES

(Presented by Academician M. A. Lavrent'ev on 4 IV 1968)

1. The classical problem of moduli of Riemann surfaces has as its starting point B. Riemann’s remark \((^{13})\) that the classes of conformally equivalent closed surfaces of genus \(g>1\) depend on \(3g-3\) complex parameters, called moduli, and consists in studying the nature of these parameters and, if possible, introducing these parameters so that they define a complex-analytic structure in the corresponding space of Riemann surfaces. In investigations of this problem by various authors it became clear that conformal equivalence of Riemann surfaces should be replaced by a weaker notion of equivalence, in which topological restrictions are imposed on the conformal mappings.

A surface of type \((g,n)\) is a surface \(S\) obtained from a closed Riemann surface of genus \(g\) by deleting \(n \geqslant 0\) distinct points. In what follows it is assumed everywhere that \(3g-3+n>0\). We shall call a Riemann surface \(S\) of type \((g,n)\), together with its canonical dissection fixed up to a homeomorphism of \(S\) onto itself homotopic to the identity, a marked Riemann surface. The consideration of homeomorphisms of one marked surface onto another is equivalent to specifying the homotopy class of homeomorphisms of the corresponding Riemann surfaces. The Teichmüller space \(T_{g,n}\) is the set of classes \([S]\) of conformally equivalent marked surfaces \(S\) of type \((g,n)\), with metric
\[ \rho\bigl([S],[S']\bigr)=\inf \log K(f), \]
where the infimum is taken over all \(K\)-quasiconformal homeomorphisms \(f:S\to S'\) with \(K=K(f)<\infty\), and \(S\) and \(S'\) are concrete representatives of the classes \([S]\) and \([S']\). For \(n=0\) the surfaces under consideration are closed.

R. Fricke (see \((^{15})\)) and O. Teichmüller (\((^{14})\), see also \((^{2,4,10})\)) proved that the space \(T_{g,n}\) is homeomorphic to the \((6g-6+2n)\)-dimensional Euclidean space \(E^{6g-6+2n}\). In the works \((^{3,5,7,9,12})\) it was proved that a complex-analytic structure can be introduced in \(T_{g,n}\). This structure turns \(T_{g,n}\) into a complex-analytic manifold of dimension \(3g-3+n\) and determines the coordinates of the points of the space \(T_{g,n}\), i.e. the moduli of the corresponding marked surfaces, only locally. L. Ahlfors \((^{1})\) and L. Bers \((^{6})\) proved that there exists a biholomorphic homeomorphism of the space \(T_{g,n}\) onto a bounded domain \(B_m \subset C^m\), \(m=3g-3+n\). The coordinates of the points of the domain \(B_m\) are global moduli of the points of the space \(T_{g,n}\). Below some elements of the construction \((^{1,6})\) will be used. We also note the work \((^{8})\), in which a global real analytic structure in \(T_{g,n}\) is defined.

1. In the present paper new moduli of marked Riemann surfaces are introduced, having a more transparent character. These moduli are equivalent to the global moduli of Ahlfors—Bers and connect the problem of moduli of Riemann surfaces with the problem of coefficients for a certain class of univalent analytic functions.

Let \(U_1\) be the disk \(|z|<1\), \(U_2\) the domain \(1<|z|\leq\infty\), and let \(\Gamma\) be a Fuchsian group of the first kind without elliptic elements, generated by a finite number of generators, for which the circle \(\gamma: |z|=1\) is a limiting circle. Then, as is known, \(S=U_1/\Gamma\) is a surface of type \((g,n)\), where \(m=3g-3+n>0\), and the surface \(\overline S=U_2/\Gamma\) is the mirror image of the surface \(S\). Denote by \(B(U_1,\Gamma)\) the complex Banach space of functions \(\varphi(z)\) analytic in the disk \(U_1\) and satisfying the condition \(\varphi(Az)A'^2(z)=\varphi(z)\) for all \(A\in\Gamma\), with norm
\[ \|\varphi\|_{B(U_1,\Gamma)}=\sup_{|z|\leq 1}(1-|z|^2)^2|\varphi(z)|<\infty. \]
By the Riemann–Roch theorem, \(\dim B(U_1,\Gamma)=m\).

Lemma. Let \(\varphi_1(z),\varphi_2(z),\ldots,\varphi_m(z)\) be a basis of the space \(B(U_1,\Gamma)\). Then the Wronskian determinant
\[ W_m(z)=\det\|\varphi_k^{(p)}(z)\|,\qquad k=1,2,\ldots,m;\quad p=0,1,\ldots,m-1, \]
on any compact set \(F\subset U_1\) can vanish only at a finite number of points.

Let \(\varphi_1(z),\varphi_2(z),\ldots,\varphi_m(z)\) be a basis of the space \(B(U_1,\Gamma)\) fixed for what follows. If \(W_m(0)=0\), then we pass to the conjugate group \(\Gamma_1=C_0\Gamma C_0^{-1}\), where \(C_0\) is a fractional-linear mapping of the disk \(U_1\) onto itself that carries the point \(z_0\), at which \(W_m(z_0)\ne0\), to zero. Therefore one may assume that \(W_m(0)\ne0\).

Let \(B_1(\Gamma)\) be the set of complex-valued functions \(\mu(z)\) measurable in the \(z\)-plane such that \(\mu(z)d\overline z/dz\) is invariant with respect to \(\Gamma\), \(\|\mu\|_{L_\infty(U_2)}<1\), and \(\mu(z)=0\) for \(z\in U_1\). For each \(\mu\in B_1(\Gamma)\) there exists a unique quasiconformal homeomorphism \(w=f_\mu(z)\) of the \(z\)-plane onto itself, satisfying the Beltrami equation \(w_{\overline z}=\mu w_z\) and normalized by the conditions \(f_\mu(0)=0\), \(f_\mu'(0)=1\), \(f_\mu(\infty)=\infty\). Then \(\Gamma^\mu=w_\mu\Gamma(w_\mu)^{-1}\) is a quasifuchsian group of fractional-linear mappings of the \(w\)-plane with invariant curve \(\gamma^\mu=f_\mu(\gamma)\), isomorphic to the group \(\Gamma\), and \(S^\mu=f_\mu(U_2)/\Gamma^\mu\) is a Riemann surface. When \(\mu\) runs through the set \(B_1(\Gamma)\), we obtain any point \([S^\mu]\), \(S^\mu=f_\mu(\overline S)\), of the Teichmüller space \(T(S)=T_{g,n}\). We shall call the functions \(\mu\) and \(\nu\) equivalent if they determine the same point of the space \(T_{g,n}\), i.e., if the surfaces \(S^\mu\) and \(S^\nu\) are conformally equivalent and the mappings \(f_\mu\) and \(f_\nu\) are homotopic. The equivalence of the functions \(\mu\) and \(\nu\) from \(B_1(\Gamma)\) is equivalent to the fact that \(f_\mu(z)=f_\nu(z)\) for \(z\in U_1\). The corresponding equivalence class \([\mu]\) of functions \(\mu\in B_2(\Gamma)\), the class of mappings \(w_\mu(z)\), will be denoted by \([w_\mu]\).

Theorem 1. Let \(w=f_\mu(z)\in[w_\mu]\) and
\[ f_\mu(z)=z+\sum_{k=2}^{\infty}a_k z^k,\qquad z\in U_1. \]
If the function
\[ g_\nu(z)=z+\sum_{k=2}^{\infty} b^k,\qquad z\in U_1 \]
is the restriction, for \(z\in U_1\), of a homeomorphism \(w=f_\nu(z)\) from some class \([w_\nu]\) and satisfies the conditions \(b_k=a_k,\ k=2,3,\ldots,m+1\), then \(f_\nu(z)=f_\mu(z)\) in \(U_1\) and \([w_\mu]=[w_\nu]\).

For the proof one considers the Schwarzian derivative \(\varphi^\mu(z)=\{f_\mu,z\}\) of the homeomorphism \(f_\mu(z)\) for \(|z|<1\). As is known \((^{11})\), \(\varphi^\mu\in B(U_1,\Gamma)\). Substituting into the equality
\[ \varphi^\mu(z)=\sum_{l=0}^{m}\omega_l\varphi_l(z),\qquad \omega_l=\mathrm{const}, \]
the Taylor expansions
\[ \varphi^\mu(z)=\sum_{k=0}^{\infty}\alpha_k z^k \quad\text{and}\quad \varphi_l(z)=\sum_{k=0}^{\infty}\alpha_{kl}z^k,\quad |z|<1, \]
and equating coefficients of like powers \(z^k,\ k=0,1,\ldots,m-1\), we obtain, for determining \(\omega_1,\ldots,\omega_m\), an algebraic system

of the equations
\[ \sum_{l=1}^{m} a_{kl}\omega_l=a_k,\qquad k=0,1,\ldots,m-1, \]
whose determinant \(\det\|a_{kl}\|\), together with \(W_m(0)\), is different from zero. Hence, noting that the coefficients \(a_0,a_1,\ldots,a_{m-1}\) are uniquely expressed in terms of \(a_2,a_3,\ldots,a_{m+1}\) by known formulas, we obtain that \(\varphi^\mu(z)\) is uniquely determined by the coefficients \(a_2,\ldots,a_{m+1}\). Consequently, \(\varphi^\mu(z)=\varphi^\nu(z)\) for \(z\in U_1\), and then, by what was proved in \((^1)\), we have \(f_\mu(z)=f_\nu(z)\) for \(z\in \overline{U}_1\), and therefore \([w_\mu]=[w_\nu]\).

The point with coordinates \((a_2,a_3,\ldots,a_{m+1})\)—the coefficients of the homeomorphisms \(f_\mu(z)\) from the fixed class \([w_\mu]\)—will be denoted by \(a_\mu\).

Theorem 2. The mapping
\[ \tau:[\mu]\to a_\mu=(a_2,a_3,\ldots,a_{m+1}) \]
is a biholomorphic homeomorphism of the space \(T_{g,n}=T(S)\) onto a bounded domain \(D_m\subset C^m\).

(Holomorphy is understood here in the sense that if \(\mu\in B_1(\Gamma)\), as an element of \(L_\infty(U_2)\), depends holomorphically on complex parameters, then \(a_2,a_3,\ldots,a_{m+1}\) also depend holomorphically on these parameters, and conversely.*) The continuity and holomorphy of the mapping \(\tau\mu=a_\mu\) is established with the aid of the variational formula
\[ w_\lambda(z)=z-\frac{z^2}{\pi}\iint_{U_2}\frac{\lambda(\zeta)\,d\sigma(\zeta)}{\zeta^2(\zeta-z)}+O(\varepsilon^2),\qquad |z|\le R<\infty, \]
where \(\varepsilon=\|\lambda\|_{L_\infty(U_2)}\) is small; this formula is applied to the mapping \(w_{\mu+\nu}(w_\mu)\), when \(\|\nu\|_{L_\infty(U_2)}\) is small. Hence, using the result mentioned above \((^1,^6)\) that the mapping \(\varkappa:[\mu]\to\varphi^\mu(z)\) is a biholomorphic homeomorphism of \(T_{g,n}\) onto a domain \(B_m\subset C^m\), by Theorem 1 we obtain that the mapping
\[ \tau\circ\varkappa^{-1}:B_m\to D_m \]
(and together with it also \(\tau\)) is a biholomorphic homeomorphism and \(D_m\) is a domain in \(C^m\).

By Theorems 1 and 2, the coordinates \((a_2,a_3,\ldots,a_{m+1})\) of the points of the domain \(D_m\) are global moduli of the points of the space \(T_{g,n}\). All known properties of the domain \(B_m\) (see \((^6)\)), invariant under biholomorphic mappings (for example, holomorphic convexity, homogeneity), are transferred also to the domain \(D_m\).

It follows from Theorems 1 and 2 that the moduli problem for Riemann surfaces is equivalent to a certain coefficient problem for the class of univalent functions under consideration \(f_\mu(z)\) for \(|z|<1\), i.e. to the problem of finding the range of values of \((a_2,a_3,\ldots,a_{m+1})\); and, since \(g\) and \(n\) may take arbitrary natural values, obtaining quantitative estimates for the boundary points of \(D_m\) is also connected with the solution of the coefficient problem in the class of all functions univalent in the disk \(|z|<1\),
\[ f(z)=z+\sum_{k=2}^{\infty}a_k z^k . \]

Institute of Mathematics
Siberian Branch of the Academy of Sciences of the USSR

Received
29 III 1968

REFERENCES

\(^1\) L. Ahlfors, Lectures on Quasiconformal Mappings, Princeton, 1966.
\(^2\) L. Ahlfors, Collected Works of L. Ahlfors and L. Bers, Spaces of Riemann Surfaces and Quasiconformal Mappings, IL, 1961, p. 104.
\(^3\) L. Ahlfors, ibid., p. 51.
\(^4\) L. Bers, ibid., p. 9.
\(^5\) L. Bers, ibid., p. 80.
\(^6\) L. Bers, On Moduli of Riemann Surfaces, Zürich, 1964.
\(^7\) A. Weil, Séminaire Bourbaki, May, 1958.
\(^8\) L. Keen, Ann. Math., 84, No. 3, 404 (1966).
\(^9\) K. Kodaira, D. S. Spencer, Ann. Math., 67, No. 2, 328 (1958).
\(^10\) S. L. Krushkal, Siberian Mathematical Journal, 8, No. 2, 313 (1967).
\(^11\) Z. Nehari, Bull. Am. Math. Soc., 55, No. 6, 545 (1949).
\(^12\) H. E. Rauch, Comm. Pure and Appl. Math., 13, 543 (1959).
\(^13\) B. Riemann, Works, Moscow–Leningrad, 1948, p. 88.
\(^14\) O. Teichmüller, Abhandl. d. Preuss. Acad. d. Wissensch., Math.-Naturwiss. Klasse, No. 4 (1943).
\(^15\) R. Fricke, F. Klein, Vorlesungen über die Theorie der automorphen Funktionen, 1, Leipzig, 1926.

* More precisely, for given \(a_2,a_3,\ldots,a_{m+1}\) there is found a certain function \(\mu\in B_1(\Gamma)\), depending holomorphically on them.