# Advances in the Theory of Riemann Surfaces. (AM-66)

LARS V. AHLFORS
LIPMAN BERS
HERSHEL M. FARKAS
ROBERT C. GUNNING
IRWIN KRA
HARRY E. RAUCH
Pages: 430
https://www.jstor.org/stable/j.ctt9qh044

1. Front Matter
(pp. i-iv)
2. PREFACE
(pp. v-vi)
The Editors
(pp. vii-x)
4. SOME REMARKS ON KLEINIAN GROUPS
(pp. 1-6)
Willaim Abikoff

In this note, we will construct four Kleinian groups. The first is finitely generated and possesses limit points which are not in the boundary of any component of the ordinary set of the group. The construction yields a counter-example to the following assertion of Lehner [1]: If G is a discontinuous group with limit set Σ and ordinary set Ω, then Σ = ∪ Bd (Ωi) where the Ωiare the components of Ω. The remaining examples are of infinitely generated groups whose limit set has positive area. Each example will show that a statement, either proved or believed to...

5. VANISHING PROPERTIES OF THETA FUNCTIONS FOR ABELIAN COVERS OF RIEMANN SURFACES (unramified case)
(pp. 7-18)
Robert D. M. Accola

1. Introduction. The vanishing properties of hyperelliptic theta functions have been known since the last century [3]. Recently, Farkas [1] discovered special vanishing properties for theta functions associated with surfaces which admit fixed-point free automorphisms of period two. The author has discovered other vanishing properties for special surfaces admitting abelian automorphism groups of low order. The purpose of this report is to give a partial exposition of a theory that will subsume most of the above cases in a general theory. Due to limitations of time and space, a full exposition must be postponed.

Let W1be a closed Riemann surface...

6. REMARKS ON THE LIMIT POINT SET OF A FINITELY GENERATED KLEINIAN GROUP
(pp. 19-26)
Lars V. Ahlfors

1. It is a conjecture of several years’ standing that the limit point set of a finitely generated Kleinian group has areal measure zero at least under some restrictive assumptions. This question has proved to be very elusive and all attempts to find the answer have been abortive. The method I am going to describe is no exception. It leads deceptively close to a solution, but my efforts to push it through have again been futile.

I am nevertheless using this opportunity to publish some of my ideas on the subject that I believe to be independently interesting.

2. We use conventional...

7. EXTREMAL QUASICONFORMAL MAPPINGS
(pp. 27-52)
Lipman Bers

This paper originated in an attempt to interpret R. S. Hamilton’s important contribution to the theory of quasiconformal mappings. Our main result (Theorem 5) is a generalization of Hamilton’s theorem [12]. It deals with the following problem.

Let D be an open set in the Riemann sphere, whose complement contains more than two points, and let G be a Kleinian group which maps D onto itself; the trivial group G = 1 is included. We consider quasiconformal automorphisms w of the Riemann sphere, such that wGw−1is again a Kleinian group. Two such mappings are called equivalent if they coincide...

8. ISOMORPHISMS BETWEEN TEICHMÜLLER SPACES
(pp. 53-80)
Lipman Bers and Leon Greenberg

In this paper we prove a theorem about isomorphisms between Teichmüller spaces (Theorem 1 below) which has been announced without proof in [5,6]. The proof, while much shorter than the original version, is still complicated. We were unable to simplify it further, so that there is no justification for delaying publication. We refer to [1, 2, 5, 6, 7, 8] for information on Teichmüller space theory. The isomorphism theorem is related to a theorem on Fuchsian groups (Theorem 2 below) which can be stated in purely classical terms. Our proofs of the two theorems are closely intertwined. Recently Marden [13]...

9. ON THE MAPPING CLASS GROUPS OF CLOSED SURFACES AS COVERING SPACES
(pp. 81-116)
Joan S. Birman and Hugh M. Hilden

Let Tgbe a closed, orientable surface of genus g, and let H(Tg) be the group of all orientation-preserving homeomorphisms of Tg→ Tg. H(Tg) contains a subgroup D(Tg) consisting of all homeomorphisms which are isotopic to the identity. The mapping class group M(Tg) of Tgis defined to be the quotient group H(Tg)/D(Tg). Alternatively, M(Tg) is known as the Teichmüller modular group, and also as the homeotopy group of Tg(although the latter term usually includes the orientation-reversing homeomorphisms of Tg→ Tg). M(Tg) can also be characterized algebraically as the group of all classes of “proper” automorphisms of...

10. SCHWARZIAN DERIVATIVES AND MAPPINGS ONTO JORDAN DOMAINS
(pp. 117-118)
Peter L. Duren
11. ON THE MODULI OF CLOSED RIEMANN SURFACES WITH SYMMETRIES
(pp. 119-130)
Clifford J. Earle

This article has two main sections. The first, §2, concerns the Teichmüller theory of closed Riemann surfaces with automorphisms. It is closely related to the author’s joint research with J. Eells (see [3] and [4]) and A. Schatz [5]. In fact, §5 of [5] treats the same situation for compact bordered surfaces. Our present treatment is a bit more thorough since we introduce the relative Teichmüller spaces T(X, H). Such spaces were first considered by A. Kuribayashi [9].

While §2 was in progress, the paper of Alling and Greenleaf on Klein surfaces and real algebraic function fields appeared [1]; it...

12. AN EIGENVALUE PROBLEM FOR RIEMANN SURFACES
(pp. 131-140)
Leon Ehrenpreis

The first part of this lecture is expository; in the second part we shall present some new ideas.

Part I.The two dimensional picture. We denote by H the upper half plane:τ= x + iy, y > 0, and we denote by G the group SL(2, R) of 2 by 2 real matrices of determinant 1. Actually, we shall be a little sloppy in in what follows in that we shall not distinguish between SL(2, R) and the factor SL(2, R)/ ± I, denoting both of them by G. We are sure that this will not cause any difficulty...

(pp. 141-156)
Hershel M. Farkas

If S is a compact Riemann surface of genus g, g ≥ 2, then the vector space of holomorphic quadratic differentials on S is a 3g−3 dimensional space. We shall denote this space by A2. If S is not hyperelliptic, then by Noether’s theorem [6], one can choose a basis for A2from the g(g+1)/2 quadratic differentialsζiζj, i ≥ j = 1, …, g whereζ1, …,ζgis a basis for the holomorphic abelian differentials on S. Choosing such a basis we can then express eachζiζjas a linear combination of the basis elements. Our...

14. DEFORMATIONS OF EMBEDDINGS OF RIEMANN SURFACES IN PROJECTIVE SPACE
(pp. 157-174)
Frederick Gardiner

Let Γ be a covering group of a compact Riemann surface of genus g where Γ operates on U, the upper half plane. Let Bq(Γ, U) be the space of holomorphic functionsϕin U which satisfy the relationϕ(γ(z))γ′(z)q=ϕ(z) for allγ ∊ Γ. Bq(Γ, U) is called the space of holomorphic q-differentials and we shall always take q to be an integer ≥ 1. The Riemann surface S = U/Γ, is mapped into P(Bq(Γ, U)*), the projective space of the dual of Bq(Γ, U), in a natural way. Namely, given pU/Γ, if p...

15. LIPSCHITZ MAPPINGS AND THE p-CAPACITY OF RINGS IN n-SPACE
(pp. 175-194)
F. W. Gehring

l. Introduction. Given a domain D in euclidean n-space Rnand given 1 ≤ p < ∞, we let Np(D) denote the collection of all continuous complex-valued functions u defined on D which are absolutely continuous in the sense of Tonelli or ACT in D with${{\left\| \text{u} \right\|}_{\text{p}}}=\underset{\text{D}}{\mathop{\sup }}\,\left| \text{u} \right|+{{\left( \int_{\text{D}}{{{\left| \nabla \text{ u} \right|}^{\text{P}}}\text{d}{{\text{m}}_{\text{n}}}} \right)}^{1/\text{p}}}<\infty \ .$

Then Np(D) is a Banach algebra under pointwise addition and multiplication with ‖ ‖pas norm. We call it theRoydenp-algebraof D.

Suppose next that D and D′ are domains in Rn. In 1960, M. Nakai [13] showed that when n = 2, the algebras Nn(D) and Nn(D′) are isomorphic if...

16. SPACES OF FUCHSIAN GROUPS AND TEICHMÜLLER THEORY
(pp. 195-204)
William. J. Harvey

The primary aim of this article is to describe an approach to the study of moduli of Fuchsian groups. The basic ideas, which have roots in the work of Fricke, Nielsen, and others, are those of Macbeath as exhibited in his papers and lectures, and many of the results are due to him. Certain simplifications of Teichmüller theory appear, in particular the simple demonstration that the fixed set in Teichmüller space of an element of the modular group is homeomorphic to a Teichmüller space, a result originally obtained by Kravetz in the classical case.

Let G be a Fuchsian group,...

17. ON FRICKE MODULI
(pp. 205-224)
Linda Keen

Let S be a closed Riemann surface of genus g from which m conformal disks have been removed. S has signature (g; m). The Teichmüller space T(S) is a topological manifold of real dimensionτ= 6g − 6 + 3m and has a natural real analytic structure. In [10], Fricke describes a set of global real analytic coordinates for this space. They are the traces of certain matrices in SL(2, R), and simple real analytic functions of them; these determine the Fuchsian group G which represents S. In this paper a set of global real analytic coordinates for T(S)...

18. EICHLER COHOMOLOGY AND THE STRUCTURE OF FINITELY GENERATED KLEINIAN GROUPS
(pp. 225-264)
Irwin Kra

There are a number of theorems on the structure of finitely generated Kleinian groups whose proofs depend on the cohomology theory introduced by Eichler [9] for Fuchsian groups. Foremost among these are Ahlfors’ finiteness theorem [1] and Bers’ area inequalities [6]. Also, the structure of these cohomology groups (Ahlfors [3], Kra [13] and [14]) is of interest, since it may reveal further information about the Kleinian groups. In this note we present an outline of this cohomology theory as developed by the author in [13] and [14] and derive from it the finiteness theorem and area inequalities. Along the way...

19. ON THE DEGENERATION OF RIEMANN SURFACES
(pp. 265-286)
Aaron Lebowitz

This paper deals with compactifying the embedding of the Torelli space in the Siegel upper half-plane [see 5]. The question that arises is, what happens if one tends to the “boundary” of Torelli space? Or, to put it another way, how do the period matrices behave?

In this paper we find results confirming one’s expectations in certain sufficiently typical types of approach to the boundary, namely, in the cases where a handle is dropped by being pinched off and where the surface splits into two of lower genera by pinching a waist. The results are enunciated in theorems one and...

20. SINGULAR RIEMANN MATRICES
(pp. 287-294)
Joseph Lewittes

In the first section of this paper we give a sufficient condition for a Riemann matrix to be singular; that is, to admit more than one principal matrix. Our main result is that the normalized period matrix of a compact Riemann surface of genus greater than one is a singular Riemann matrix if the surface admits an automorphism of order two, other than the hyperelliptic involution. Then a connection is pointed out between these singular period matrices and a new class of theta functions on the surface. In the second section we illustrate some of the theory by a concrete...

21. AN INEQUALITY FOR KLEINIAN GROUPS
(pp. 295-296)
Albert Marden

The purpose of this note is to point out that an elementary 3-dimensional topological argument yields the following result.

Theorem. Suppose G is a Kleinian group with N generators, Ω is its set of ordinary points, and Ω/G = ∪ Siis the decomposition of Ω/G into its components. If giis the genus of Sithen

Σ gi≤ N .

Corollary. If G is purely loxodromic then Ω/G has at most N/2 components.

This corollary, which is an immediate consequence of Ahlfors’ finiteness theorem [1] and of the preceding theorem, is apparently not a consequence of the deep...

22. ON KLEIN’S COMBINATION THEOREM III
(pp. 297-316)

The basic idea of Klein’s combination theorem is as follows. One is given two Kleinian groups G1and G2satisfying certain algebraic and geometric conditions. One concludes that G, the group generated by G1and G2is again Kleinian, one can find a fundamental set for G, and the algebraic structure of G is determined.

In Klein’s original combination theorem [4], G is the free product of G1and G2. In the first paper of this series [7], G is the free product of G1and G2with an amalgamated cyclic subgroup. In the second paper [8], G2is cyclic,...

23. ON FINSLER GEOMETRY AND APPLICATIONS TO TEICHMÜLLER SPACES
(pp. 317-328)
Brian O’Byrne

In this paper we investigate the following situation in Finsler geometry: (X,α) and (Y,β) are complete Finsler manifolds such that there exists a C1+-foliation f: X → Y and the Finsler structureβis, in a sense to be defined later, the infimum via f of the Finsler structureα. It is possible to define two metrics on Y which are related toα. One is the metric induced byβand the other is, again in a sense to be defined later, the infimum via f of the metric induced on X byα. Our main result...

24. REPRODUCING FORMULAS FOR POINCARÉ SERIES OF DIMENSION −2 AND APPLICATIONS
(pp. 329-340)
K. V. Rajeswara Rao

Let Γ denote a group of conformal self-maps of the open unit disc U of the complex plane, acting discontinuously and freely on U. Thus Γ is to contain no elliptic transformations. Our concern is with the Hilbert space A(Γ) of square integrable automorphic forms of dimension −2 belonging to Γ, or, equivalently, with holomorphic differentials on the Riemann surface U/Γ which are square integrable in the sense of Ahlfors and Sario ([1], Ch. V).

We try to construct elements of A(Γ) as Poincaré series of dimension −2 : ΣTΓf(Tz)·T′(z). For the existence of such Poincaré series, the natural...

25. PERIOD RELATIONS ON RIEMANN SURFACES
(pp. 341-354)
Harry E. Rauch

In recent joint work, [4], Farkas and I established explicit period relations on a compact Riemann surface of genus g ≥ 4, relations which include Schottky’s relation, [11] and [10], for g = 4 (the first such relation known), and, indeed, give what we regard as the first satisfactory proof of it, and which for g > 4 are the direct structural generalizations of it first conjectured rather imprecisely by Schottky and Jung, [12]. For convenience I call such period relationsrelations of Schottky type. Very succinctly, given a Riemann surface S of genus g ≥ 4 and on it a...

26. SCHOTTKY IMPLIES POINCARÉ
(pp. 355-364)
Harry E. Rauch

In this note I show that (one form of) Schottky’s period relation for Riemann surfaces of genus four, [7] and [6], implies Poincaré’a approximate period relation, [3] (cf. also [2]), for surfaces of genus four whose period matrices are close to diagonal form.

Namely:

\caption {1)}\begin{align*} & \surd \theta [\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ \end{matrix}]\theta [\begin{matrix} 0 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ \end{matrix}]\theta [\begin{matrix} 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 \\ \end{matrix}]\theta [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 \\ \end{matrix}]\times \\ & \surd \theta [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{matrix}]\theta [\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 1 \\ \end{matrix}]\theta [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ \end{matrix}]\theta [\begin{matrix} 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ \end{matrix}]\pm \\ & \surd \theta [\begin{matrix} 1 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{matrix}]\theta [\begin{matrix} 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ \end{matrix}]\theta [\begin{matrix} 1 & 0 & 0 & 1 \\ 1 & 1 & 1 & 1 \\ \end{matrix}]\theta [\begin{matrix} 1 & 1 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ \end{matrix}]\times \\ & \surd \theta [\begin{matrix} 0 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ \end{matrix}]\theta [\begin{matrix} 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ \end{matrix}]\theta [\begin{matrix} 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ \end{matrix}]\theta [\begin{matrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ \end{matrix}]\pm \\ & \surd \theta [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ \end{matrix}]\theta [\begin{matrix} 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ \end{matrix}]\theta [\begin{matrix} 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 \\ \end{matrix}]\theta [\begin{matrix} 0 & 0 & 1 & 1 \\ 0 & 1 & 1 & 1 \\ \end{matrix}]\times \\ & \surd \theta [\begin{matrix} 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ \end{matrix}]\theta [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ \end{matrix}]\theta [\begin{matrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{matrix}]\theta [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 \\ \end{matrix}]=0 \\ \end{align*}

implies, I claim,

\caption {2)}\begin{align*} & \sqrt{{{\pi }_{12}}\ {{\pi }_{23}}\ {{\pi }_{34}}\ {{\pi }_{14}}+0({{\epsilon }\ ^{10}})}\ \pm \\ & \sqrt{{{\pi }_{14}}\ {{\pi }_{24}}\ {{\pi }_{23}}\ {{\pi }_{13}}+0({{\epsilon }\ ^{10}})}\ \pm \\ & \sqrt{{{\pi }_{13}}\ {{\pi }_{34}}\ {{\pi }_{24}}\ {{\pi }_{12}}+0({{\epsilon }\ ^{10}})}=0\ , \\ \end{align*}

where= max √|πij| over all 1 ≤ i < j ≤ 4. It must be understood that I am dealing here with a Riemann surface S of genus four and on it a canonical homology basisγ1,… ,γ4;δ1,… ,δ4and the set dζ1,… ,dζ4of...

27. TEICHMÜLLER MAPPINGS WHICH KEEP THE BOUNDARY POINTWISE FIXED
(pp. 365-368)
Edgar Reich and Kurt Strebel

Given the measurable functionμ(z), U: {|z| < 1},$\underset{\text{U}}{\mathop{\text{ess sup}}}\,\left| \mu (z) \right|<1$

let fμdenote the quasiconformal homeomorphism of U onto U with complex dilatationμ, normalized such that fμ(1) = 1, fμ(i) = i, fμ(−1) = −1· Our purpose is to study conditions onμfor which the boundary correspondence induced by fμis the identity. This abstract contains a summary of some of the results. For proofs, further results, applications, and bibliography the reader is referred to [1] and [2].

Let F = {μ|fμ(eiθ) ≡ eiθ, 0 ≤θ< 2π}.

Theorem 1. Ifμ ∊Fthe following condition is necessary:...

28. AUTOMORPHISMS AND ISOMETRIES OF TEICHMÜLLER SPACE
(pp. 369-384)
H. L. Royden

1. The Teichmüller space Tg. Let W be a compact surface of genus g ≥ 2, and let W0be W with a fixed complex analytic structure. Then every complex analytic structure on W is specified by giving a Beltrami differentialμon W0. The Riemann surface given byμis denoted by Wμ. Two structuresμandνare said to be conformally equivalent if there is a conformal mapping of Wμonto Wν, and they are said to be equivalent if this mapping can be taken to be homotopic to the identity. The Teichmüller space Tgof Riemann surfaces...

29. DEFORMATIONS OF EMBEDDED RIEMANN SURFACES
(pp. 385-392)
Reto A. Rüedy

It was Felix Klein who realized that each surface embedded in 3-space which is sufficiently smooth (we always assume C) can be viewed as a Riemann surface in a natural way, namely by declaring a parameter admissible if and only if the corresponding mapping is conformal. The existence of such parameters is a deep but well-known fact (see [2], pp. 15-41). We will call Riemann surfaces of this kindclassicalRiemann surfaces.

Polyhedrons embedded in 3-space can be viewed as Riemann surfaces in the same way, but here the existence of admissible parameters is almost trivial: We triangulate the polyhedron...

30. FOCK REPRESENTATIONS AND THETA-FUNCTIONS
(pp. 393-406)
Ichiro Satake

This lecture is mostly of expository nature. Our purpose is to explain some basic facts on Fock representations along with their connection to theta-functions. A particular emphasis will be placed on the analogy between Fock representations and discrete series representations of Harish-Chandra. We will observe Fock spaces as members of a family of(mutually equivalent) representation-spaces parametrized by a Siegel upper half-space and give an explicit form of the integral operator intertwining the representations on these spaces. In this way, we will obtain simple proofs to a result of Shale and Weil ([7], [8.2]) on the existence of a projective unitary...

31. ‘UNIFORMIZATIONS’ OF INFINITELY CONNECTED DOMAINS
(pp. 407-421)
R. J. Sibner

1. Let D be a family of Riemann surfaces and S a subfamily. In the theory of conformal mappings, what might be called the “universal uniformization problem,” is to show the existence of an element SS in each isomorphism [i.e. conformal equivalence] class of D.

For example, if D is the family of Riemann surfaces of genus zero and D the subfamily of plane domains, the above is Koebe’s “general uniformization principle.” For D the family of simply connected surfaces and S the subfamily consisting of the disk, the finite plane and the extended plane, it is the Riemann...

32. Back Matter
(pp. 422-422)