Covering properties of open continuous mappings having two valences between Riemann surfaces

1995 ◽  
Vol 118 (2) ◽  
pp. 321-340 ◽  
Author(s):  
Abdallah Lyzzaik
Keyword(s):  
Riemann Surface ◽  
Finite Number ◽  
Riemann Surfaces ◽  
Embedding Theorem ◽  
Branch Points ◽  
Continuous Mappings ◽  
Counting Multiplicity ◽  
Covering Properties ◽  
Closed Riemann Surface ◽  
Finite Genus

AbstractLet be an open Riemann surface with finite genus and finite number of boundary components, and let be a closed Riemann surface. An open continuous function from to is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in either p or q times, counting multiplicity, with possibly a finite number of exceptions. These comprise the most general class of all non-trivial functions having two valences between and .The object of this paper is to study the geometry of (p, q)-maps and establish a generalized embedding theorem which asserts that the image surfaces of (p, q)-maps embed in p-fold closed coverings possibly having branch points off the image surfaces.

The geometry of open continuous mappings having two valences between Riemann surfaces

1996 ◽  
Vol 120 (2) ◽  
pp. 309-329 ◽  
Author(s):  
Abdallah Lyzzaik
Keyword(s):  
Riemann Surface ◽  
Finite Number ◽  
Riemann Surfaces ◽  
Branch Points ◽  
Continuous Mappings ◽  
Counting Multiplicity ◽  
Covering Properties ◽  
Finite Set ◽  
Closed Riemann Surface ◽  
Finite Genus

AbstractAn open continuous function from an open Riemann surface with finite genus and finite number of boundary components into a closed Riemann surface is termed a (p, q)-map, 0 < q < p, if it has a finite number of branch points and assumes every point in the image surface either p or q times, counting multiplicity, with possibly a finite number of exceptions.The object of this paper is to prove that the geometry of any (p, q)-map resembles that of a (p, q)-map whose q-set (the set of image points of f that are taken on exactly q times, counting multiplicity), constitutes a finite set of Jordan arcs or curves (loops). This leads to interesting geometrie results regarding (p, q)-maps without exceptional points. Further, it yields that every (p, q)-map is homotopic to a simplicial (p, q)-map having the same covering properties.

Invariant Subspaces on Riemann Surfaces

1966 ◽  
Vol 18 ◽  
pp. 399-403 ◽  
Author(s):  
Michael Voichick
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Finite Number ◽  
Harmonic Measure ◽  
Riemann Surfaces ◽  
Invariant Subspaces ◽  
Riemann Surface With Boundary ◽  
Analytic Curves

In this paper we generalize to Riemann surfaces a theorem of Helson and Lowdenslager in (2) describing the closed subspaces of L2(﹛|z| = 1﹜) that are invariant under multiplication by eiθ.Let R be a region on a Riemann surface with boundary Γ consisting of a finite number of disjoint simple closed analytic curves such that R ⋃ Γ is compact and R lies on one side of Γ. Let dμ be the harmonic measure on Γ with respect to a fixed point t0 on R. We shall consider the closed subspaces of L2(Γ, dμ) that are invariant under multiplication by functions in A (R) = ﹛F|F continuous on , analytic on R}.

scholarly journals Schottky uniformizations of closed Riemann surfaces with Abelian groups of conformal automorphisms

1994 ◽  
Vol 36 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Rubén A. Hidalgo
Keyword(s):  
Abelian Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Abelian Groups ◽  
Its Region ◽  
Schottky Group ◽  
Schottky Groups ◽  
Schottky Uniformization ◽  
Closed Riemann Surface ◽  
Holomorphic Covering

Let us consider a pair (S, H) consisting of a closed Riemann surface S and an Abelian group H of conformal automorphisms of S. We are interested in finding uniformizations of S, via Schottky groups, which reflect the action of the group H. A Schottky uniformization of a closed Riemann surface S is a triple (Ώ, G, π:Ώ→S) where G is a Schottky group with Ώ as its region ofdiscontinuity and π:Ώ→S is a holomorphic covering with G ascovering group. We look for a Schottky uniformization (Ώ, G, π:Ώ→S) of S such that for each transformation h in H there exists an automorphisms t of Ώ satisfying h ∘ π = π ∘ t.

scholarly journals EQUISYMMETRIC STRATA OF THE MODULI SPACE OF CYCLIC TRIGONAL RIEMANN SURFACES OF GENUS 4

2009 ◽  
Vol 51 (1) ◽  
pp. 19-29 ◽  
Author(s):  
MILAGROS IZQUIERDO ◽  
DANIEL YING
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Fuchsian Groups ◽  
Regular Covering ◽  
Closed Riemann Surface

AbstractA closed Riemann surface which can be realized as a three-sheeted covering of the Riemann sphere is called trigonal, and such a covering is called a trigonal morphism. If the trigonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic trigonal Riemann surface. Using the characterization of cyclic trigonality by Fuchsian groups, we find the structure of the space of cyclic trigonal Riemann surfaces of genus 4.

scholarly journals On the Hyperelliptic Riemann Surfaces of Infinite Genus with Absolutely Convergent Riemann’s Theta Functions

1968 ◽  
Vol 33 ◽  
pp. 57-73 ◽  
Author(s):  
Kenichi Tahara
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Theta Functions ◽  
Hyperelliptic Riemann Surfaces ◽  
Hyperelliptic Riemann Surface ◽  
Closed Riemann Surface

The Riemann’s theta functions associated with a closed Riemann surface are absolutely convergent. In the present paper, we shall show an example of an hyperelliptic Riemann surface of infinite genus such that the Riemann’s theta functions associated with are absolutely convergent.

scholarly journals SCHOTTKY UNIFORMIZATIONS OF SYMMETRIES

2013 ◽  
Vol 55 (3) ◽  
pp. 591-613 ◽  
Author(s):  
G. GROMADZKI ◽  
R. A. HIDALGO
Keyword(s):  
Riemann Surface ◽  
Kleinian Group ◽  
Riemann Surfaces ◽  
Algebraic Curve ◽  
Schottky Group ◽  
Structural Description ◽  
Schottky Groups ◽  
Real Algebraic Curve ◽  
Schottky Uniformization ◽  
Closed Riemann Surface

AbstractA real algebraic curve of genus g is a pair (S,〈 τ 〉), where S is a closed Riemann surface of genus g and τ: S → S is a symmetry, that is, an anti-conformal involution. A Schottky uniformization of (S,〈 τ 〉) is a tuple (Ω,Γ,P:Ω → S), where Γ is a Schottky group with region of discontinuity Ω and P:Ω → S is a regular holomorphic cover map with Γ as its deck group, so that there exists an extended Möbius transformation $\widehat{\tau}$ keeping Ω invariant with P o $\widehat{\tau}$=τ o P. The extended Kleinian group K=〈 Γ, $\widehat{\tau}$〉 is called an extended Schottky groups of rank g. The interest on Schottky uniformizations rely on the fact that they provide the lowest uniformizations of closed Riemann surfaces. In this paper we obtain a structural picture of extended Schottky groups in terms of Klein–Maskit's combination theorems and some basic extended Schottky groups. We also provide some insight of the structural picture in terms of the group of automorphisms of S which are reflected by the Schottky uniformization. As a consequence of our structural description of extended Schottky groups, we get alternative proofs to results due to Kalliongis and McCullough (J. Kalliongis and D. McCullough, Orientation-reversing involutions on handlebodies, Trans. Math. Soc. 348(5) (1996), 1739–1755) on orientation-reversing involutions on handlebodies.

STRING THEORIES ON RIEMANN SURFACES OF ALGEBRAIC FUNCTIONS

1990 ◽  
Vol 05 (14) ◽  
pp. 2799-2820 ◽  
Author(s):  
FRANCO FERRARI
Keyword(s):  
Abelian Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Scalar Fields ◽  
Conformal Weight ◽  
Energy Tensor ◽  
Branch Points ◽  
Stress Energy Tensor ◽  
Tensor Method ◽  
Stress Energy

In this paper we extend to a general Riemann surface a formalism used so far for surfaces with Abelian group of symmetry. Using an algebraic equation F(z, w)=0 to define the surface in terms of sheets and branch points, it is possible to construct the correlation functions for b—c systems with integer conformal weight j and for the scalar fields X. Explicit examples are provided for a general surface of genus three and a surface of genus four. No essential complication arises with respect to the hyperelliptic case. At the end we discuss the computation of chiral determinants det [Formula: see text] using the stress energy tensor method.

Extremal lengths of homology classes on Riemann surfaces

1999 ◽  
Vol 1999 (508) ◽  
pp. 17-45
Author(s):  
Makoto Masumoto
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Conformal Invariant ◽  
Compact Riemann Surfaces ◽  
Finite Genus

Abstract We assign a positive number V, a new conformal invariant, to a Riemann surface R of finite genus in terms of the extremal lengths of certain weak homology classes on R, and determine the range of V. In particular we find algebraic relations among the extremal lengths of homology classes on compact Riemann surfaces.

ON THE PARTITION FUNCTION OF STRING THEORIES DEFINED ON ALGEBRAIC CURVES

1992 ◽  
Vol 07 (21) ◽  
pp. 5131-5154 ◽  
Author(s):  
FRANCO FERRARI
Keyword(s):  
Riemann Surface ◽  
String Theory ◽  
Partition Function ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Bosonic String ◽  
Branch Points ◽  
Bosonic String Theory ◽  
String Theories

In this paper the amplitudes of bosonic string theory on Riemann surfaces are studied taking the branch points as moduli. The case of a general Riemann surface of genus three is completely worked out, constructing the chiral determinants and the propagators. The chiral determinants and the partition function are given explicitly in terms of the moduli and of the parameters of the curve apart from a factor which is essentially a theta constant. The Beilinson-Manin formulae for Riemann surfaces whose moduli space is parametrized by branch points are discussed.

scholarly journals On real trigonal Riemann surfaces

2006 ◽  
Vol 98 (1) ◽  
pp. 53 ◽  
Author(s):  
Antonio F. Costa ◽  
Milagros Izquierdo
Keyword(s):  
Fixed Point ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Riemann Sphere ◽  
Connected Components ◽  
Klein Surface ◽  
Fixed Point Set ◽  
Regular Covering ◽  
Closed Riemann Surface ◽  
Point Set

A closed Riemann surface $X$ which can be realized as a 3-sheeted covering of the Riemann sphere is called trigonal, and such a covering will be called a trigonal morphism. A trigonal Riemann surface $X$ is called real trigonal if there is an anticonformal involution (symmetry) $\sigma$ of $X$ commuting with the trigonal morphism. If the trigonal morphism is a cyclic regular covering the Riemann surface is called real cyclic trigonal. The species of the symmetry $\sigma $ is the number of connected components of the fixed point set $\mathrm{Fix}(\sigma)$ and the orientability of the Klein surface $X/\langle\sigma\rangle$. We characterize real trigonality by means of Fuchsian and NEC groups. Using this approach we obtain all possible species for the symmetry of real cyclic trigonal and real non-cyclic trigonal Riemann surfaces.

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