scholarly journals On Principal Function Problem

1970 ◽  
Vol 38 ◽  
pp. 85-90 ◽  
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Boundary Behavior ◽  
Ideal Boundary ◽  
Principal Function ◽  
Open Riemann Surface ◽  
The Given ◽  
The Ideal

Sario’s theory of principal functions fully discussed in his research monograph [3] with Rodin stems from the principal function problem which is to find a harmonic function p on an open Riemann surface R imitating the ideal boundary behavior of the given harmonic function s in a neighborhood A of the ideal boundary δ of R.

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′.

scholarly journals Radon-Nikodym Densities between Harmonic Measures on the Ideal Boundary of an Open Riemann Surface

1966 ◽  
Vol 27 (1) ◽  
pp. 71-76
Author(s):  
Mitsuru Nakai
Keyword(s):  
Riemann Surface ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Compact Set ◽  
Ideal Boundary ◽  
Subharmonic Functions ◽  
Open Riemann Surface ◽  
Harmonic Measures ◽  
The Ideal

Resolutive compactification and harmonic measures. Let R be an open Riemann surface. A compact Hausdorff space R* containing R as its dense subspace is called a compactification of R and the compact set Δ = R* -R is called an ideal boundary of R. Hereafter we always assume that R does not belong to the class OG. Given a real-valued function f on Δ, we denote by the totality of lower bounded superharmonic (resp. upper bounded subharmonic) functions sonis satisfying

scholarly journals Functions meromorphic on some Riemann surfaces

1978 ◽  
Vol 70 ◽  
pp. 41-45
Author(s):  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Ideal Boundary ◽  
Open Riemann Surface ◽  
A Value ◽  
Extended Complex ◽  
The Ideal ◽  
Extended Complex Plane

Consider an open Riemann surface R and a single-valued meromorphic function w = f(p) defined on R. A value w0 in the extended complex plane is said to be a cluster value for w = f(p) if there exists a sequence {pn } in R accumulating at the ideal boundary of R such that

scholarly journals A Riemann on the ideal boundary of a Riemann surface

1956 ◽  
Vol 32 (6) ◽  
pp. 409-411 ◽  
Author(s):  
Shin'ichi Mori ◽  
Minoru Ota
Keyword(s):  
Riemann Surface ◽  
Ideal Boundary ◽  
The Ideal

scholarly journals P-harmonic dimensions on ends

1993 ◽  
Vol 132 ◽  
pp. 131-139
Author(s):  
Michihiko Kawamura ◽  
Shigeo Segawa
Keyword(s):  
Riemann Surface ◽  
Boundary Component ◽  
Harmonic Functions ◽  
Ideal Boundary ◽  
Boundary Values ◽  
Hölder Continuous ◽  
Closed Curves ◽  
Open Riemann Surface ◽  
Relative Boundary ◽  
Bounded Harmonic Functions

Consider an end Ω in the sense of Heins (cf. Heins [3]): Ω is a relatively non-compact subregion of an open Riemann surface such that the relative boundary ∂Ω consists of finitely many analytic Jordan closed curves, there exist no non-constant bounded harmonic functions with vanishing boundary values on ∂Ω and Ω has a single ideal boundary component. A density P = P(z)dxdy (z = x + iy) is a 2-form on Ω∩∂Ω with nonnegative locally Holder continuous coefficient P(z).

The parabolicity of Brelot's harmonic spaces

1995 ◽  
Vol 59 (1) ◽  
pp. 20-27
Author(s):  
Hideo Imai
Keyword(s):  
Harmonic Function ◽  
Ideal Boundary ◽  
Minimal Growth ◽  
Positive Harmonic Function ◽  
The Ideal

AbstractThe parabolicity of Brelot's harmonic spaces is characterized by the fact that every positive harmonic function is of minimal growth at the ideal boundary.

scholarly journals Parabolic Stein Manifolds

2014 ◽  
Vol 114 (1) ◽  
pp. 86 ◽  
Author(s):  
A. Aytuna ◽  
A. Sadullaev
Keyword(s):  
Riemann Surface ◽  
Subharmonic Function ◽  
Arbitrary Dimension ◽  
Topological Type ◽  
Zero Set ◽  
Space Of Analytic Functions ◽  
Exhaustion Function ◽  
Stein Manifolds ◽  
Open Riemann Surface ◽  
The Given

An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the first part of this note we compile these notions of parabolicity and give some immediate relations among these different definitions. In section 3 we relate some of these notions to the linear topological type of the Fréchet space of analytic functions on the given manifold. In section 4 we look at some examples and show, for example, that the complement of the zero set of a Weierstrass polynomial possesses a continuous plurisubharmonic exhaustion function that is maximal off a compact subset.

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.

Sign in / Sign up

Export Citation Format

Share Document