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 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.

A soft Oka principle for proper holomorphic embeddings of open Riemann surfaces into (ℂ*)2

2018 ◽  
Vol 2018 (745) ◽  
pp. 59-82 ◽  
Author(s):  
Tyson Ritter
Keyword(s):  
Riemann Surface ◽  
Recent Result ◽  
Riemann Surfaces ◽  
Complex Structure ◽  
Open Riemann Surface ◽  
Oka Principle ◽  
Continuous Maps ◽  
Holomorphic Embeddings ◽  
The Given ◽  
Smooth Deformation

Abstract Let X be an open Riemann surface. We prove an Oka property on the approximation and interpolation of continuous maps X \to (\mathbb{C}^{*})^{2} by proper holomorphic embeddings, provided that we permit a smooth deformation of the complex structure on X outside a certain set. This generalises and strengthens a recent result of Alarcón and López. We also give a Forstnerič–Wold theorem for proper holomorphic embeddings (with respect to the given complex structure) of certain open Riemann surfaces into {(\mathbb{C}^{*})^{2}} .

PRESCRIBED HORIZONTAL AND VERTICAL TREES PROBLEM OF QUADRATIC DIFFERENTIALS

2006 ◽  
Vol 08 (03) ◽  
pp. 381-399
Author(s):  
THOMAS KWOK-KEUNG AU ◽  
TOM YAU-HENG WAN
Keyword(s):  
Riemann Surface ◽  
Quadratic Differential ◽  
Quadratic Differentials ◽  
Sufficient Condition ◽  
Holomorphic Quadratic Differential ◽  
Simply Connected ◽  
The Given

A sufficient condition for the existence of holomorphic quadratic differential on a non-compact simply-connected Riemann surface with prescribed horizontal and vertical trees is obtained. In particular, for any pair of complete ℝ-trees of finite vertices with (n + 2) infinite edges, there exists a polynomial quadratic differential on ℂ of degree n such that the associated vertical and horizontal trees are isometric to the given pair.

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 INDUCED QUANTUM GRAVITY ON A RIEMANN SURFACE

2003 ◽  
Vol 18 (24) ◽  
pp. 4371-4401 ◽  
Author(s):  
G. BANDELLONI ◽  
S. LAZZARINI
Keyword(s):  
Riemann Surface ◽  
Simple Model ◽  
Quantum Gravity ◽  
Arbitrary Dimension ◽  
Perturbative Expansion ◽  
Target Space ◽  
Quantum Corrections ◽  
Classical Formula ◽  
Local Coordinates ◽  
Nontrivial Character

Induced quantum gravity dynamics built over a Riemann surface is studied in arbitrary dimension. Local coordinates on the target space are given by means of the Laguerre–Forsyth construction. A simple model is proposed and perturbatively quantized. In doing so, the classical [Formula: see text]-symmetry turns out to be preserved on-shell at any order of the ℏ perturbative expansion. As a main result, due to quantum corrections, the target coordinates acquire a nontrivial character.

scholarly journals On the induced distribution of the shape of the projection of a randomly rotated configuration

2010 ◽  
Vol 42 (02) ◽  
pp. 331-346
Author(s):  
H. Le ◽  
D. Barden
Keyword(s):  
Euclidean Space ◽  
Arbitrary Dimension ◽  
Shape Space ◽  
Size And Shape ◽  
The Given

Using the geometry of the Kendall shape space, in this paper we study the shape, as well as the size-and-shape, of the projection of a configuration after it has been rotated and, when the given configuration lies in a Euclidean space of an arbitrary dimension, we obtain expressions for the induced distributions of such shapes when the rotation is uniformly distributed.

scholarly journals Boundary Isomorphism between Dirichlet Finite Solutions of Δu = Pu and Harmonic Functions

1973 ◽  
Vol 50 ◽  
pp. 7-20 ◽  
Author(s):  
Ivan J. Singer
Keyword(s):  
Differential Equation ◽  
Riemann Surface ◽  
Laplace Operator ◽  
Nontrivial Solution ◽  
Harmonic Functions ◽  
Green’S Functions ◽  
Hölder Continuous ◽  
Open Riemann Surface ◽  
Continuous Density ◽  
The Laplace Operator

Consider an open Riemann surface R and a density P(z)dxdy (z = x + iy), well defined on R. As was shown by Myrberg in [3], if P ≢ 0 is a nonnegative α-Hölder continuous density on R (0 < α ≤ 1) then there exists the Green’s functions of the differential equationp>on R, where Δ means the Laplace operator. As a consequence, there always exists a nontrivial solution on R.

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).

scholarly journals Computation of the topological type of a real Riemann surface

2014 ◽  
Vol 83 (288) ◽  
pp. 1823-1846 ◽  
Author(s):  
C. Kalla ◽  
C. Klein
Keyword(s):  
Riemann Surface ◽  
Topological Type
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