scholarly journals Abel’s theorem for analytic mappings of an open Riemann surface into compact Riemann surfaces of genus one

1978 ◽  
Vol 18 (2) ◽  
pp. 305-325 ◽  
Author(s):  
Masakazu Shiba
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Compact Riemann Surfaces ◽  
Genus One ◽  
Analytic Mappings

scholarly journals Analytic Functions on Some Riemann Surfaces

1963 ◽  
Vol 22 ◽  
pp. 211-217 ◽  
Author(s):  
Nobushige Toda ◽  
Kikuji Matsumoto
Keyword(s):  
Riemann Surface ◽  
Compact Subset ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Interesting Theorem ◽  
Hyperbolic Riemann Surface

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

Rings of Meromorphic Functions on Non-Compact Riemann Surfaces

1969 ◽  
Vol 21 ◽  
pp. 284-300 ◽  
Author(s):  
James Kelleher
Keyword(s):  
Riemann Surface ◽  
Algebraic Structure ◽  
Riemann Surfaces ◽  
Meromorphic Functions ◽  
Doctoral Dissertation ◽  
University Of Illinois ◽  
Compact Riemann Surfaces ◽  
The University ◽  
Complex Valued

In this paper we shall be concerned with the algebraic structure of certain rings of functions meromorphic on a non-compact (connected) Riemann surface Ω. In this setting, A = A(Ω) and K= K(Ω) denote (respectively) the ring of all complex-valued functions analytic on Ω and its field of quotients, the field of functions meromorphic on Ω. The rings considered here are those subrings of K containing A,which we term A-rings of K. Most of the results given here were previously announced without proof (15) and are contained in the author's doctoral dissertation (16), completed at the University of Illinois under the direction of Professor M. Heins, whose encouragement and advice are gratefully acknowledged.

scholarly journals Open Riemann Surface with Null Boundary

1951 ◽  
Vol 3 ◽  
pp. 73-79 ◽  
Author(s):  
Kiyoshi Noshiro
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Open Riemann Surface ◽  
Capacity Zero

Recently the writer has obtained some results concerning meromorphic or algebroidal functions with the set of essential singularities of capacity zero, with an aid of a theorem of Evans. In the present paper, suggested from recent interesting papers of Sario and Pfluger, the writer will extend his results to single-valued analytic functions defined on open abstract Riemann surfaces with null boundary in the sense of Nevanlinna, using a lemma instead of Evans’ theorem.

scholarly journals Some Theorems on Open Riemann Surfaces

1951 ◽  
Vol 3 ◽  
pp. 141-145 ◽  
Author(s):  
Masatsugu Tsuji
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Green’S Function ◽  
Green's Function ◽  
Riemann Surfaces ◽  
Dirichlet Integral ◽  
Open Riemann Surface ◽  
Surface Spread

Let F be an open Riemann surface spread over the z-plane. We say that F is of positive or null boundary, according as there exists a Green’s function on F or not, Let u(z) be a harmonic function on Fand be its Dirichlet integral As R. Nevanlinna proved, if F is of null boundary, there exists no one-valued non-constant harmonic function on F5 whose Dirichlet integral is finite, This Nevanlinna’s theorem was proved very simply by Kuroda.

On a Uniqueness Theorem

1979 ◽  
Vol 31 (5) ◽  
pp. 1072-1076
Author(s):  
Mikio Niimura
Keyword(s):  
Riemann Surface ◽  
Uniqueness Theorem ◽  
Analytic Functions ◽  
Unit Disc ◽  
Riemann Surfaces ◽  
Boundary Behavior ◽  
Uniqueness Theorems ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
Hyperbolic Riemann Surfaces

The classical uniqueness theorems of Riesz and Koebe show an important characteristic of boundary behavior of analytic functions in the unit disc. The purpose of this note is to discuss these uniqueness theorems on hyperbolic Riemann surfaces. It will be necessary to give additional hypotheses because Riemann surfaces can be very nasty. So, in this note the Wiener compactification will be used as ideal boundary of Riemann surfaces. The Theorem, Corollaries 1, 2 and 3 are of Riesz type, Riesz-Nevanlinna type, Koebe type and Koebe-Nevanlinna type respectively. Corollaries 4 and 5 are general forms of Corollaries 2 and 3 respectively.Let f be a mapping from an open Riemann surface R into a Riemann surface R′.

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().

ON THE CONNECTEDNESS OF THE LOCUS OF REAL ELLIPTIC-HYPERELLIPTIC RIEMANN SURFACES

2009 ◽  
Vol 20 (08) ◽  
pp. 1069-1080 ◽  
Author(s):  
JOSÉ A. BUJALANCE ◽  
ANTONIO F. COSTA ◽  
ANA M. PORTO
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Orbit Space ◽  
Previous Article ◽  
Connected Components ◽  
Hyperelliptic Involution ◽  
Hyperelliptic Surfaces ◽  
Genus One

A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace [Formula: see text] of real elliptic-hyperelliptic algebraic curves in the moduli space [Formula: see text] of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space [Formula: see text] of real elliptic-hyperelliptic surfaces of genus > 5: we show that [Formula: see text] is connected if g is odd and has exactly two connected components if g is even; in both cases the closure [Formula: see text] of [Formula: see text] in the compactified moduli space [Formula: see text] is connected.

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]).

scholarly journals The Riemann surface of a ring

1969 ◽  
Vol 10 (1-2) ◽  
pp. 251-256
Author(s):  
M. G. Stanley
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Contravariant Functor ◽  
Trivial Representation ◽  
Analytic Mapping ◽  
Algebra Of Functions ◽  
The Right ◽  
Analytic Mappings

The contravariant functor F from the category of Riemann surfaces and analytic mappings to the category of complex algebras and homomorphisms which takes each surface Ω to the algebra of analytic functions on Ω does not have an adjoint on the right; but it nearly does. To each algebra A there is associated a surface Σ1 (A) and a homomorphism A from A into FΣ1 (A), indeed onto an algebra of functions not all of which are constant on any component of Σ1 (A), such that every such non-trivial representation A A → F(Ω) is induced by a unique analytic mapping Ω → Σ1(A)

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