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.

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 Embedded minimal surfaces of finite topology

2019 ◽  
Vol 2019 (753) ◽  
pp. 159-191 ◽  
Author(s):  
William H. Meeks III ◽  
Joaquín Pérez
Keyword(s):  
Asymptotic Behavior ◽  
Riemann Surface ◽  
Finite Number ◽  
Minimal Surface ◽  
Minimal Surfaces ◽  
Compact Riemann Surface ◽  
Finite Topology ◽  
Compact Boundary ◽  
Interior Points

AbstractIn this paper we prove that a complete, embedded minimal surface M in {\mathbb{R}^{3}} with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface {\overline{M}} with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion {\overline{M}}. In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior of M.

scholarly journals Complete Classification of Degree 7 for Genus 1

2021 ◽  
pp. 594-603
Author(s):  
Peshawa M. Khudhur
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Symmetric Group ◽  
Compact Riemann Surface ◽  
Complete Classification ◽  
Conjugacy Classes ◽  
Branched Cover ◽  
Transitive Subgroup ◽  
Genus One

Assume that  is a meromorphic fuction of degree n where X is compact Riemann surface of genus g. The meromorphic function gives a branched cover of the compact Riemann surface X. Classes of such covers are in one to one correspondence with conjugacy classes of r-tuples (  of permutations in the symmetric group , in which  and s generate a transitive subgroup G of  This work is a contribution to the classification of all primitive groups of degree 7, where X is of genus one.

scholarly journals Applications of Extremal Length to Classification of Riemann Surfaces

1964 ◽  
Vol 24 ◽  
pp. 159-166
Author(s):  
Tatsuo Fuji’i’e
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Extremal Length ◽  
Connected Component ◽  
Countable Number ◽  
Relative Boundary ◽  
Analytic Curves

Let D be a subregion of a Riemann surface F, whose relative boundary consists of at most countable number of analytic curves which do not cluster in F. For a regular exhaustion {Fn} of F, we put Dn = D∩ (F— Fn), and define the extremal radius R(P, ∂Dn) of the relative boundary ∂Dn of Dn, measured at a point P(∈F0) of F with respect to the connected component of F - Dn which contains P. Let K(|z|≦r) be a disk centered at P and contained in a parametric disk of 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]).

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.

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