Theory Of Riemann Surfaces Research Articles

Algorithms for the computation of polynomial relationships for the hard hexagon model

We calculate the polynomial relationships P( ϱ, z) = 0 and P( ϱ, κ) = 0 in both the ordered and disordered regimes for the hard hexagon model. We start with Baxter's exact solution which gives the physical quantities κ, ϱ, and z parametrically as a function of a variable τ and exploit the modular properties of Baxter's solution. Using elementary Riemann surface theory, the computation can be reduced to algorithms involving only linear algebra. These algorithms are implemented using a computer algebra system. The method will be applicable to other exactly solvable models in which cusp expansions can be computed.

Harmonic morphisms and circle actions on 3- and 4-manifolds

Harmonic morphisms are considered as a natural generalization of the analytic functions of Riemann surface theory. It is shown that any closed analytic 3-manifold supporting a non-constant harmonic morphism into a Riemann surface must be a Seifert fibre space. Harmonic morphisms φ:M→N from a closed 4-manifold to a 3-manifold are studied. These determine a locally smooth circle action on M with possible fixed points. This restricts the topology of M. In all cases, a harmonic morphism φ:M→N from a closed (n+1)-dimensional manifold to an n-dimensional manifold (n≥2, with M, N analytic in the case n=2) determines a locally smooth circle action on M.

FUNCTIONAL INTEGRAL METHODS FOR FERMIONIC STRING AND SUPER RIEMANN SURFACES

We investigate the light-cone gauge formulation of fermionic string of Mandelstam in a superspace. To formulate the fermionic string in a superspace, we use the theory of super Riemann surfaces (SRS). We define the Neumann functions and the Mandelstam mappings in a superspace by means of the so-called Abelian differentials of the first and the third kinds on SRS. In the super Schottky parametrization of SRS these superdifferentials are constructed at the multiloop level. The functional integral measure of a super light-cone diagram, which consists of 6g−6+2N even moduli parameters and 4g−4+2N odd ones, are specified in our formulation. In one-loop case with even spin structures we explicitly evaluate the superdeterminants on the super light-cone diagrams and calculate the one-loop amplitudes with N massless vector states. It is shown that the result is written in the form of a correlation function of the vertex operators. Furthermore, we evaluate the 3- and 4-particle amplitudes explicitly, which agree exactly with those calculated in the Green-Schwarz formulation.

The Behnke-Stein theorem for open Riemann surfaces

Using the Riemann-Roch theorem and the set-topological part of Bishop's polyhedron lemma, we show that the usual Runge approximation theorem for compact subsets of the Riemann sphere is valid word-for-word on any compact Riemann surface X, with meromorphic functions on X playing the role of rational functions; this result is essentially equivalent to the Behnke-Stein approximation theorem. The Behnke-Stein generalization of the Runge approximation theorem [1], which is the basic tool for many existence questions on open Riemann surfaces, can be stated in several equivalent ways, for instance: Let X be a Riemann surface, and K a compact subset of X. Then, in order that every holomorphic function in a neighborhood of K be uniformly approximable on K by holomorphic functions on X, it is necessary and sufficient that X K have no connected component with compact closure in X. As is well known, the necessity part of the above theorem follows from the sufficiency part (using the theory of compact Riemann surfaces). In this note, we wish to point out how the famous special polyhedron lemma of Bishop [2], which is an elementary set-topological result, can be used (together with the Riemann-Roch theorem for compact Riemann surfaces) to give a simple proof of the following theorem. Theorem 1.1. Let X be a compact Riemann surface, and K c X a compact subset. Let Q be any subset of X K which contains (precisely) one point qi from each connected component W of X K. Then any holomorphic function on a neighborhood of K can be approximated uniformly on K by meromorphic functions on X whose poles lie in Q. Again it is well known and easy to see that Theorem 1.1 implies the sufficiency part of Behnke-Stein. The main fact needed for this implication is the following: Received by the editors March 24, 1988. 1980 Mathematics Subject Classification (1 985 Revision). Primary 30E 10, 30F10.

Automorphic functions on super Riemann surfaces and fermionic string

Using the theory of super Riemann surfaces, the light-cone gauge formulation of the fermionic string by Mandelstam is extended to the one-loop level. The one-loop N-massless vector particle amplitude is explicitly calculated. The higher-loop extension is also discussed. We also demonstrate the equivalence between the light-cone picture and the Polyakov picture at the one-loop level.

The Riemann boundary value problem on closed Riemann surfaces and integrable systems

The Riemann problem has different forms. In several areas of physics (scattering theory, integrable systems and so on) it is necessary to build an analytic function or a matrix with prescribed poles and zeros on a Riemann surface. Another, similar, problem is to construct a section of a complex vector bundle over a Riemann surface (for example, anti-self-dual Yang-Mills equation). As was established by A. Grothendieck and H. Röhrl, this problem is equivalent to the Riemann boundary value problem: let M be a compact Riemann surface, Γ a contour on M, G(p) an n × n matrix on Γ; determine an analytic matrix F(p) in MΓ which is multiple of a given divisor ∗⧹γ and satisfies the boundary condition on Γ F + ( p) = G( p) F −( p). This problem is well understood on the complex plane. It is related to singular integral operators, Wiener-Hopf equations and Banach algebras. In the case of a Riemann surface there are additional with the Riemann-Roch and Abel theorems, Jacobi varieties and Riemann theta-functions. These relations are described here. The Riemann problem is equivalent to the so-called ∂ -problem ∂u ∂ λ = Au . Recently this elliptic equation was used to study the inverse scattering problem. Finally, we consider some physical problems (finite-gap potentials, the Landau-Lifschitz equation, the Painlevé problems) to illustrate the applications of the Riemann problem on a Riemann surface. Acquaintance with Riemann surface theory and algebraic topology is not assumed.

Measured foliations and the minimal norm property for quadratic differentials

The objective is to prove two important inequalities of Riemann surface theory and to present them in such a way that one can see their close relationship. The first is the minimal norm property of quadratic differentials first observed by Marden and Strebel [8]. Actually a somewhat different is given. The minimum norm properties have many applications, one of which is a proof of the result of Hubbard and Masur [7] Concerning the existence of a unique holomorphic quadratic differential whose horizontal foliation and vertical measure realize a given measure class of measured foliations on a given Riemann surface. This result is given in section 7. The method of proof is quite different from the one in [7]. The second important is the of Reich and Strebel. It has central importance in Teichmiiller theory and can be used to derive the infinitesimal form of Teichmiiller's metric [10]. Moreover, this can be used to show that Teichmiiller's metric is the integral of its infinitesimal form [6]. A proof of this result by other methods was given by O'Byrne in [9]. Both of these inequalities turn out to be consequences of certain properties of the trajectory structure of a holomorphic quadratic differential. For the purposes of the main inequality we are able to bypass some of the theory of trajectories by means of an averaging device used by Teichmiiller. This averaging device also appears in the proofs of Teichmiiller's theorem given by Bers [4] and Abikoff [1]. But for the minimum norm property in the form given by Marden and Strebel [8], the detailed theory of trajectories seems to be necessary.

Quadrature identities and the schottky double

By using Riemann surface theory we obtain results on quadrature domains and identities for analytic functions, e.g., existence of multiply-connected quadrature domains, descriptions of their algebraic boundaries and results on the multitude of quadrature domains associated to a fixed quadrature identity. The main idea is to characterize quadrature domains in terms of meromorphic functions and differentials on Riemann surfaces conformally equivalent to the Schottky doubles of the domains.

A note on the generalized inverse Nyquist stability criterion

The purpose of this paper is to provide a self-contained proof of the generalized inverse Nyquist stability criterion which is based on the theory of algebraic functions and Riemann surfaces. It is shown how the fixed modes are related to the polynomial determining the inverse characteristic gains.

A simple proof of Radó’s theorem

In this note we show that Rado's theorem [3] (sometimes called the Rado-BehnkeStein-Cartan theorem-see [2] and the literature cited there) is an easy consequence of a basic lemma in the analytic theory of Riemann surfaces, often called Weyl's lemma [1]. Rad6's theorem reads: If f(z) is a complex-valued continuous on the unit disc D and analytic where it is not zero, then it is analytic in all of D. Weyl's lemma states: If X is a real-valued Lebesgue measurable on D such that f fD 0 * AA dx dy = 0 for every C? ,u with compact support contained in D, then X is almost everywhere equal to a harmonic on D. Here is how to deduce Rado's theorem from Weyl's lemma. Let f(z) = u(z) + iv(z); we shall use Weyl's lemma to show that u and v are harmonic on all of D. Let D = Z U N where Z = {z E D: u(z) = 0} and N is the open subset of D where u(z) is nonzero and, hence, is harmonic. Let {Ui} be a countable locally finite open covering of N by disc-like sets, and let {ei(z)} be an associated C?' partition of unity. Finally, let ,u(z) be a C' test function as above. We have

On the generalized Nyquist stability criterion

The purpose of this paper is to provide a self-contained proof of the generalized Nyquist stability criterion. The frequency-dependent eigenvalues of a square transfer-function matrix G(8) are defined directly from a matrix fraction decomposition of G(8), and a discussion of the fixed modes and fixed gains is given. The theory of algebraic functions and Riemann surfaces is employed in the proof, and a detailed consideration is given to these theoretical ideas from the point of view of the stability result. The question of the relationship between the poles and zeros of an algebraic function (in the complex-variable sense) and the poles and zeros (in the Smith-McMillan sense) is carefully examined, and a proof of the argument principle for a regular region on a Riemann surface is given.

On relative Teichmüller spaces and a globalization principle in Riemann surface theory

On relative Teichmüller spaces and a globalization principle in Riemann surface theory

The modular symbol and continued fractions in higher dimensions

The modular symbol has proved a very useful tool for dealing with holomorphic automorphic forms on the upper half-plane. It serves, for instance, to compute the action of Hecke operators on automorphic forms, to give the value of twisted L-functions of certain elliptic curves in terms of a finite sum of modular symbols, and to prove that divisors of degree zero supported on the cusps of a modular curve have finite order in the Jacobian of that curve. For these matters, see [1], Chaps. IV and V, and the references cited there. Here is a definition of the modular symbol. Let H be the upper half-plane, that is, the set of all complex numbers with positive imaginary part. A cusp of H is defined to be any rational number (considered as a point of the real line) or oc. The cusps belong to the boundary of H in the extended complex plane. Let SL(2,1R) act on H in the usual way. For any subgroup F of finite index in SL(2, 9 ) we can form the quotient H/F. There is a closed, compact Riemann surface X(F) such that H/F is biholomorphic to the complement of some finite set of points x 1 . . . . ,x~ in X(F). These points x i are called the cusps of X(F). Let ~?(F) denote the holomorphic differential l-forms on X(F). Integration of forms over cycles yields a canonical N-linear map from H~(X(F);IR) to the complex dual space of ~(F). The theory of Riemann surfaces tells us that this map is an isomorphism. Now choose any two rational cusps a and b on the boundary of H. Let l be any reasonable arc beginning at a, ending at b, and staying in H in the interim for instance, l could be the geodesic in H from a to b. Then one easily shows that integration over the image of l in X(F) gives a linear map from ~?(F) to I12. This linear map gives an element of HI(X(F); IR) in the canonical way described above. We denote this homology class by [a, hi. We call [ ] the modular symbol. In summary, the "classical" modular symbol is a map from the set of pairs of cusps of H into H~(X(F);IR). This symbol enjoys properties which are the

A Canonical Form for Symmetric and Skew-Symmetric Extended Symplectic Modular Matrices With Applications to Riemann Surface Theory

The (extended) symplectic modular group $({ \wedge _n}){\Gamma _n}$ is the set of all $2n \times 2n$ integer matrices $M$ such that \[ (M{J^t}M = \pm J),M{J^t}M = J,J = \left [ {\begin {array}{*{20}{c}} 0 & I \\ { - I} & 0 \\ \end {array} } \right ],\] $I$ being the $n \times n$ identity matrix. Let ${S_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = - {}^tM\}$ and ${T_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = {}^tM\}$. We say $M \sim N$ if there exists $K \in {\Gamma _n}$ such that $M = KN{}^tK$. This defines an equivalence relation on each of these sets separately and we obtain a canonical form for this equivalence. We use this canonical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorphic self-map of order 2. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.

A canonical form for symmetric and skew-symmetric extended symplectic modular matrices with applications to Riemann surface theory

The (extended) symplectic modular group ( ∧ n ) Γ n ({ \wedge _n}){\Gamma _n} is the set of all 2 n × 2 n 2n \times 2n integer matrices M M such that \[ ( M J t M = ± J ) , M J t M = J , J = [ 0 a m p ; I − I a m p ; 0 ] , (M{J^t}M = \pm J),M{J^t}M = J,J = \left [ {\begin {array}{*{20}{c}} 0 & I \\ { - I} & 0 \\ \end {array} } \right ], \] I I being the n × n n \times n identity matrix. Let S n = { M ∈ ∧ n − Γ n | M = − t M } {S_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = - {}^tM\} and T n = { M ∈ ∧ n − Γ n | M = t M } {T_n} = \{ M \in { \wedge _n} - {\Gamma _n}|M = {}^tM\} . We say M ∼ N M \sim N if there exists K ∈ Γ n K \in {\Gamma _n} such that M = K N t K M = KN{}^tK . This defines an equivalence relation on each of these sets separately and we obtain a canonical form for this equivalence. We use this canonical form to study two types of Riemann surfaces which are conformally equivalent to their conjugates and obtain characterizations of their period matrices. We also obtain characterizations of the symplectic matrices which the conformal equivalence induces on the first homology group. One type of surface dealt with is the symmetric Riemann surfaces, i.e. those surfaces which have a conjugate holomorphic self-map of order 2. The other type of surface studied we we call pseudo-symmetric surfaces. These are the hyperelliptic surfaces with the property that the sheet interchange is the square of a conjugate holomorphic automorphism.

Square integrable differentials on Riemann surfaces and quasiconformal mappings

If $f:R \to R’$ is a quasiconformal homeomorphism of Riemann surfaces, then Marden showed that $f$ naturally induces an isomorphism of the corresponding Hilbert spaces of square integrable first-order differential forms. It is demonstrated that this isomorphism preserves many important subspaces. Preliminary to this, various facts about the subspace of semiexact differentials are derived; especially, the orthogonal complement is identified. The norm of the isomorphism is $K(f)^{1/2}$ where $K(f)$ is the maximal dilatation of $f$. In addition, $f$ defines an isomorphism of the square integrable harmonic differentials and some important subspaces are preserved. It is shown that not all important subspaces are preserved. The relationship of this to other work is investigated; in particular, the connection with the work of Nakai on the isomorphism of Royden algebras induced by a quasiconformal mapping is explored. Finally, the induced isomorphisms are applied to the classification theory of Riemann surfaces to show that various types of degeneracy are quasiconformally invariant.

Characters of free groups represented in the two‐dimensional special linear group

We consider here the problem of determining when two elements in a free group will have the same character under all possible representations of the given group in the special linear group of 2 x 2 matrices with determinant 1. Problems involving the representation of free groups in terms of 2 x 2 matrices with real entries and determinant 1 have been investigated by R. Fricke in connection with certain problems in the theory of Riemann surfaces [2]. Fricke’s main concern is to find those representations which define discontinuous groups of the conformal self-mappings of the upper half-plane. His method is based on arguments in non-Euclidean geometry, and has only recently been made fully rigorous and transparent by Linda Keen [4]. Apart from the analytic problems inherent in Fricke’s method there are several algebraic problems which have been answered by Fricke either incompletely or not at all. It is the algebraic aspects of Fricke’s theory which we shall investigate here. Independently of Fricke’s problems however, the subsequent development may be considered as a straightforward contribution to the representation theory of free groups. We shall consider the following general situation. K will denote a fixed commutative ring with identity of characteristic zero, and SL(2, K) the group of all 2 x 2 matrices with determinant 1 and entries from K. It will be shown that the character of any element in a free group under an arbitrary representation of the group in SL(2, K) can be constructively represented as a polynomial expression with integer coefficients in the characters of certain simple products of the group generators. Two polynomials will represent the same character if and only if they are congruent modulo a member of a certain class of ideals. This will reduce the problem of determining whether two elements have the same character to the problem of determining the structure of these ideals. The structure of the ideals will then be determined for the free groups of rank 1, 2, and for the free group of rank 3 under restricted conditions on K. We shall show that if two elements u, v in a free group F have the same character under all possible * The work for this paper was done while the author was at the Courant Institute of Mathematical Sciences and was supported by the National Science Foundation Grant GP-28538. Reproduction in whole or in part is permitted for any purpose of the United States Government. 635

Extremal length and harmonic functions on Riemann surfaces

Expressions for several conformally invariant pseudometrics on a Riemann surface R R are given in terms of three new forms of reduced extremal distance. The pseudometrics are defined by means of various subclasses of the set of all harmonic functions on R R having finite Dirichlet integral. The reduced extremal distance between two points is defined on R R , on the Alexandroff one-point compactification of R R and on the Kerékjártó-Stoïlow compactification of R R . These reduced extremal distances are computed in terms of harmonic functions having specified singularities and boundary behavior. The key to establishing this connection with harmonic functions is a general theorem dealing with extremal length on a compact bordered Riemann surface and its extensions to noncompact bordered surfaces. These results are used to obtain new tests for degeneracy in the classification theory of Riemann surfaces. Finally, some of the results are illustrated for a hyperbolic simply connected Riemann surface.

A theorem on permutations

Out of the classical theory of Riemann surfaces, we extract the following combinatorial theorem: If π 1,…, π m are permutations of 1, 2,…, n, such that π 1… π m = 1, then v( π 1) + … + v( π m ) ⩾ 2( n − t), where v( π) = n − r, r being the number of orbits of the cyclic group generated by the permutation π acting on the set {1,…, n}, and where t is the number of orbits of the group generated by π 1,…, π m . The direct proof of this theorem seems to be difficult.

A Maximum Principle for Dirichlet-Finite Harmonic Functions on Riemannian Spaces

Representations of harmonic functions by means of integrals taken over the harmonic boundary ΔR of a Riemann surface R enable one to study the classification theory of Riemann surfaces in terms of topological properties of ΔR (cf. [6; 4; 1; 7]). In deducing such integral representations, essential use is made of the fact that the functions in question attain their maxima and minima on ΔR.The corresponding maximum principle in higher dimensions was discussed for bounded harmonic functions in [3]. In the present paper we consider Dirichlet-finite harmonic functions. We shall show that every such function on a subregion G of a Riemannian N-space R attains its maximum and minimum on the set , where ∂G is the relative boundary of G in R and the closures are taken in Royden's compactification R*. As an application we obtain the harmonic decomposition theorem relative to a compact subset K of R* with a smooth ∂(K ∩ R).