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)

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.).

scholarly journals On Level Curves of Harmonic and Analytic Functions on Riemann Surfaces

1969 ◽  
Vol 34 ◽  
pp. 77-87
Author(s):  
Shinji Yamashitad
Keyword(s):  
Riemann Surface ◽  
Subharmonic Function ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Vector Lattice ◽  
Level Curves ◽  
Harmonic Majorant ◽  
Harmonic And Analytic Functions ◽  
Hyperbolic Riemann Surface

In this note we shall denote by R a hyperbolic Riemann surface. Let HP′(R) be the totality of harmonic functions u on R such that every subharmonic function | u | has a harmonic majorant on R. The class HP′(R) forms a vector lattice under the lattice operations:

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.

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′.

Invariant Subspace Theorems for Finite Riemann Surfaces

1966 ◽  
Vol 18 ◽  
pp. 240-255 ◽  
Author(s):  
Morisuke Hasumi
Keyword(s):  
Riemann Surface ◽  
Invariant Subspace ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Invariant Subspaces ◽  
Multiply Connected Domains ◽  
Disk Algebra ◽  
Multiply Connected ◽  
Beurling Theorem ◽  
Finite Riemann Surface

The purpose of this paper is to extend various invariant subspace theorems for the circle group to multiply connected domains. Such attempts are not new. Actually, Sarason (4) studied the invariant subspaces of annulus operators acting on L2 and showed certain parallelisms between the unit disk case and the annulus case. Voichick (8) observed analytic functions on a finite Riemann surface and generalized the Beurling theorem on the closed invariant subspaces of H2 as well as the Beurling–Rudin theorem on the closed ideals of the disk algebra. Here we shall consider LP(Γ) and C(Γ) defined on the boundary Γ of a finite orientable Riemann surface R.

scholarly journals The value distribution of harmonic mappings between Riemannian n-spaces

1975 ◽  
Vol 59 ◽  
pp. 45-58
Author(s):  
Hideo Imai
Keyword(s):  
Riemann Surface ◽  
Kernel Function ◽  
Riemann Surfaces ◽  
Distribution Theory ◽  
Value Distribution ◽  
Harmonic Mappings ◽  
Value Distribution Theory ◽  
Theoretic Method ◽  
Analytic Mapping ◽  
Crucial Part

We are concerned with the value distribution of a mapping of an open Riemannian n-space (n ≧ 3) into a Riemannian n-space. The value distribution theory of an analytic mapping of Riemann surfaces was initiated by S. S. Chern [1] and developed mainly by L. Sario [8], [9], [10], [11], and then by H. Wu [14], [15]. The most crucial part in Sario’s theory is the introduction of a kernel function on an arbitrary Riemann surface to describe appropriately the proximity of two points. His method indicates that the potential theoretic method is one of the powerful methods in the value distribution theory.

scholarly journals A Theorem on an Analytic Mapping of Riemann Surfaces

1961 ◽  
Vol 19 ◽  
pp. 149-157
Author(s):  
Minoru Kurita
Keyword(s):  
Differential Geometry ◽  
Riemann Surfaces ◽  
View Point ◽  
Analytic Mapping ◽  
Circular Points ◽  
Analytic Mappings

Recently S. S. Chern [1] intended an aproach to some problems about analytic mappings of Riemann surfaces from a view-point of differential geometry. In that line we treat here orders of circular points of analytic mappings. The author expresses his thanks to Prof. K. Noshiro for his kind advices.

scholarly journals On Some Families of Analytic Functions on Riemann Surfaces

1968 ◽  
Vol 31 ◽  
pp. 57-68 ◽  
Author(s):  
Shinji Yamashita
Keyword(s):  
Riemann Surface ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Harmonic Majorant

Throughout this paper all functions are single-valued. Let R be a Riemann surface. We shall denote by φ∧ the least harmonic majorant of a function φ defined in R if it has the meaning.

Range Sets And Bmo Norms of Analytic Functions

1984 ◽  
Vol 36 (4) ◽  
pp. 747-755 ◽  
Author(s):  
Shoji Kobayashi
Keyword(s):  
Riemann Surface ◽  
Analytic Function ◽  
Green's Function ◽  
Analytic Functions ◽  
Riemann Surfaces ◽  
Bounded Mean Oscillation ◽  
Open Riemann Surface ◽  
Mean Oscillation ◽  
Image Position ◽  
Dirichlet Integrals

In this paper we are concerned with the space BMOA of analytic functions of bounded mean oscillation for Riemann surfaces, and it is shown that for any analytic function on a Riemann surface the area of its range set bounds the square of its BMO norm, from which it is seen as an immediate corollary that the space BMOA includes the space AD of analytic functions with finite Dirichlet integrals.Let R be an open Riemann surface which possesses a Green's function, i.e., R ∉ OG, and f b e an analytic function defined on R. The Dirichletintegral DR(f) = D(f) of f on R is defined by1.1and we denote by AD(R) the space of all functions f analytic on R for which D(f) < +∞.

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