scholarly journals Schiffer Comparison Operators and Approximations on Riemann Surfaces Bordered by Quasicircles

2020 ◽  
Author(s):  
Eric Schippers ◽  
Mohammad Shirazi ◽  
Wolfgang Staubach
Keyword(s):  
Riemann Surface ◽  
Bergman Space ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Dirichlet Space ◽  
Holomorphic Functions ◽  
Finite Collection ◽  
Approximation Theorems ◽  
Arbitrary Genus ◽  
Simply Connected

Abstract We consider a compact Riemann surface R of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate R into two subsets: a connected Riemann surface $$\Sigma $$ Σ , and the union $$\mathcal {O}$$ O of a finite collection of simply connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $$\mathcal {O}$$ O to the Bergman space of holomorphic forms on $$\Sigma $$ Σ is an isomorphism onto the exact one-forms, when restricted to the orthogonal complement of the set of forms on all of R. We then apply this to prove versions of the Plemelj–Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some approximation theorems for the Bergman space of one-forms and Dirichlet space of holomorphic functions on $$\Sigma $$ Σ by elements of Bergman space and Dirichlet space on fixed regions in R containing $$\Sigma $$ Σ .

scholarly journals Remarks on the Elliptic Case of the Mapping Theorem for Simply-Connected Riemann Surfaces

1955 ◽  
Vol 9 ◽  
pp. 17-20 ◽  
Author(s):  
Maurice Heins
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Riemann Surfaces ◽  
Simple Pole ◽  
Conformal Map ◽  
Elliptic Case ◽  
Extended Plane ◽  
Simply Connected ◽  
Conformal Equivalence

It is well-known that the conformal equivalence of a compact simply-connected Riemann surface to the extended plane is readily established once it is shown that given a local uniformizer t(p) which carries a given point p0 of the surface into 0, there exists a function u harmonic on the surface save at p0 which admits near p0 a representation of the form(α complex 0; h harmonic at p0). For the monodromy theorem then implies the existence of a meromorphic function on the surface whose real part is u. Such a meromorphic function has a simple pole at p0 and elsewhere is analytic. It defines a univalent conformal map of the surface onto the extended plane.

scholarly journals Topological Classification of Conformal Actions onpq-Hyperelliptic Riemann Surfaces

2007 ◽  
Vol 2007 ◽  
pp. 1-29 ◽  
Author(s):  
Ewa Tyszkowska
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Topological Classification ◽  
Hyperelliptic Riemann Surfaces

A compact Riemann surfaceXof genusg>1is said to bep-hyperellipticifXadmits a conformal involutionρ, for whichX/ρis an orbifold of genusp. If in additionXisq-hyperelliptic, then we say thatXispq-hyperelliptic. Here we study conformal actions onpq-hyperelliptic Riemann surfaces with centralp- andq-hyperelliptic involutions.

Qp Spaces on Riemann Surfaces

1998 ◽  
Vol 50 (3) ◽  
pp. 449-464 ◽  
Author(s):  
Rauno Aulaskari ◽  
Yuzan He ◽  
Juha Ristioja ◽  
Ruhan Zhao
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Bloch Space ◽  
Dirichlet Space ◽  
Function Spaces ◽  
Hyperbolic Riemann Surfaces ◽  
Qp Spaces

AbstractWe study the function spaces Qp(R) defined on a Riemann surface R, which were earlier introduced in the unit disk of the complex plane. The nesting property Qp(R) ⊆Qq(R) for 0 < p < q < ∞ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space AD(R) ⊆ Qp(R) for any p, 0 < p < ∞, thus sharpening T. Metzger's well-known result AD(R) ⊆ BMOA(R). Also the first author's result AD(R) ⊆ VMOA(R) for a regular Riemann surface R is sharpened by showing that, in fact, AD(R) ⊆ Qp,0(R) for all p, 0 < p < ∞. The relationships between Qp(R) and various generalizations of the Bloch space on R are considered. Finally we show that Qp(R) is a Banach space for 0 < p < ∞.

On Rings with a Certain Type of Factorization and Compact Riemann Surfaces

1990 ◽  
Vol 42 (6) ◽  
pp. 1041-1052
Author(s):  
Pascual Cutillas Ripoll
Keyword(s):  
Riemann Surface ◽  
Finite Subset ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Meromorphic Functions ◽  
Irreducible Elements ◽  
Finite Sums ◽  
Compact Riemann Surfaces

AbstractLet be a compact Riemann surface, be the complement of a nonvoid finite subset of and A() be the ring of finite sums of meromorphic functions in with finite divisor. In this paper it is proved that every nonzero f ∈ A() can be decomposed as a product αβ, where α is either a unit or a product of powers of irreducible elements of A(), uniquely determined by f up to multiplication by units, and β is a product of functions of the type eφ – 1, with φ holomorphic and nonconstant in . Furthermore, a similar result is obtained for a certain class of subrings of A().

scholarly journals Conformal immersions of compact Riemann surfaces into the 2n-sphere (n ≥ 2)

1996 ◽  
Vol 141 ◽  
pp. 79-105 ◽  
Author(s):  
Jun-Ichi Hano
Keyword(s):  
Riemann Surface ◽  
Positive Integer ◽  
Euclidean Space ◽  
Unit Sphere ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Minimal Immersion ◽  
Dimensional Euclidean Space ◽  
Compact Riemann Surfaces

The purpose of this article is to prove the following theorem:Let n be a positive integer larger than or equal to 2, and let S be the unit sphere in the 2n + 1 dimensional Euclidean space. Given a compact Riemann surface, we can always find a conformal and minimal immersion of the surface into S whose image is not lying in any 2n — 1 dimensional hyperplane.This is a partial generalization of the result by R. L. Bryant. In this papers, he demonstrates the existence of a conformal and minimal immersion of a compact Riemann surface into S2n, which is generically 1:1, when n = 2 ([2]) and n = 3 ([1]).

BOSONIZATION AND FERMION VERTICES ON AN ARBITRARY GENUS RIEMANN SURFACE BY USING A GLOBAL OPERATOR FORMALISM

1989 ◽  
Vol 04 (24) ◽  
pp. 2349-2362 ◽  
Author(s):  
JORGE RUSSO
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Explicit Computation ◽  
Operator Formalism ◽  
Central Extensions ◽  
Arbitrary Topology ◽  
Global Operator ◽  
Arbitrary Genus ◽  
Higher Genus ◽  
Spin Fields

Fermi-Bose equivalence is studied with the use of a global operator formalism on Riemann surfaces of arbitrary topology. The quantization of a scalar field on a circle is performed in detail, globally, at arbitrary genus. A new algebra of the Krichever-Novikov type naturally emerges. This admits three central extensions and generalizes standard algebras of the sphere to higher genus. It is shown by explicit computation that the central terms, as well as correlation functions, corresponding to the Bose and Fermi models agree. Spin fields and fermion vertices are defined within this framework and their conformal properties are investigated.

Harmonic Morphisms as Sphere Bundles Over Compact Riemann Surfaces

1997 ◽  
Vol 08 (07) ◽  
pp. 935-942
Author(s):  
Sigmundur Gudmundsson
Keyword(s):  
Riemann Surface ◽  
Homotopy Type ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Harmonic Morphisms ◽  
Sphere Bundle ◽  
Harmonic Morphism ◽  
Totally Geodesic ◽  
Compact Riemann Surfaces

We prove that the projection map of an orientable sphere bundle, over a compact Riemann surface, of any homotopy type can be realized as a harmonic morphism with totally geodesic fibres.

scholarly journals Minimum topological genus of compact bordered Klein surfaces admitting a prime-power automorphism

1995 ◽  
Vol 37 (2) ◽  
pp. 221-232 ◽  
Author(s):  
E. Bujalance ◽  
J. M. Gamboa ◽  
C. Maclachlan
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Prime Power ◽  
Compact Riemann Surface ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Abelian Case ◽  
Power Automorphism ◽  
Topological Genus

In the nineteenth century, Hurwitz [8] and Wiman [14] obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, [11], [12]. These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey [7] and Maclachlan [10], respectively, in the case of Riemann surfaces and by Bujalance [2], Hall [6] and Gromadzki [5] in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf [1])—was minimized by Bujalance- Etayo-Gamboa-Martens [3] in the case where G is cyclic and by McCullough [13] in the abelian case.

Riemann Surfaces, Coverings, and Hypergeometric Functions

2017 ◽  
Author(s):  
Paula Tretkoff
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hypergeometric Functions ◽  
Euler Number ◽  
Linear Transformations ◽  
Triangle Groups ◽  
Finite Subgroups ◽  
Basic Facts ◽  
Biholomorphic Map ◽  
Simply Connected

This chapter deals with Riemann surfaces, coverings, and hypergeometric functions. It first considers the genus and Euler number of a Riemann surface before discussing Möbius transformations and notes that an automorphism of a Riemann surface is a biholomorphic map of the Riemann surface onto itself. It then describes a Riemannian metric and the Gauss-Bonnet theorem, which can be interpreted as a relation between the Gaussian curvature of a compact Riemann surface X and its Euler characteristic. It also examines the behavior of the Euler number under finite covering, along with finite subgroups of the group of fractional linear transformations PSL(2, C). Finally, it presents some basic facts about the classical Gauss hypergeometric functions of one complex variable, triangle groups acting discontinuously on one of the simply connected Riemann surfaces, and the hypergeometric monodromy group.

Dynamics of holomorphic correspondences on Riemann surfaces

2020 ◽  
Vol 31 (05) ◽  
pp. 2050036 ◽  
Author(s):  
Tien-Cuong Dinh ◽  
Lucas Kaufmann ◽  
Hao Wu
Keyword(s):  
Riemann Surface ◽  
Topological Degree ◽  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Critical Values ◽  
Probability Measures ◽  
Global Dynamics ◽  
Forward Orbit ◽  
Holomorphic Correspondences

We study the global dynamics of holomorphic correspondences [Formula: see text] on a compact Riemann surface [Formula: see text] in the case, so far not well understood, where [Formula: see text] and [Formula: see text] have the same topological degree. In the absence of a mild and necessary obstruction that we call weak modularity, [Formula: see text] admits two canonical probability measures [Formula: see text] and [Formula: see text] which are invariant by [Formula: see text] and [Formula: see text] respectively. If the critical values of [Formula: see text] (respectively, [Formula: see text]) are not periodic, the backward (respectively, forward) orbit of any point [Formula: see text] equidistributes towards [Formula: see text] (respectively, [Formula: see text]), uniformly in [Formula: see text] and exponentially fast.

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