scholarly journals Harmonic functions and quadratic harmonic morphisms on Walker spaces

2016 ◽  
Vol 40 ◽  
pp. 1004-1019 ◽  
Author(s):  
Cornelia-Livia BEJAN ◽  
Simona-Luiza DRUTA-ROMANIUC
Keyword(s):  
Harmonic Functions ◽  
Harmonic Morphisms

scholarly journals Harmonic Morphisms Projecting Harmonic Functions to Harmonic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-8
Author(s):  
M. T. Mustafa
Keyword(s):  
Harmonic Function ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Horizontal Component ◽  
Harmonic Morphisms

For Riemannian manifoldsMandN, admitting a submersionϕwith compact fibres, we introduce the projection of a function via its decomposition into horizontal and vertical components. By comparing the Laplacians onMandN, we determine conditions under which a harmonic function onU=ϕ−1(V)⊂Mprojects down, via its horizontal component, to a harmonic function onV⊂N.

HARMONIC MORPHISMS FROM EINSTEIN 4-MANIFOLDS TO RIEMANN SURFACES

2003 ◽  
Vol 14 (03) ◽  
pp. 327-337 ◽  
Author(s):  
MARINA VILLE
Keyword(s):  
Riemann Surface ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Riemann Surfaces ◽  
Complex Structure ◽  
Hermitian Structure ◽  
Harmonic Morphisms ◽  
Harmonic Morphism ◽  
The Real ◽  
Real Analyticity

If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from [Formula: see text] or [Formula: see text] to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation. We conclude by an example of a non-constant harmonic morphism from [Formula: see text] to [Formula: see text].

scholarly journals PSEUDO HARMONIC MORPHISMS ON RIEMANNIAN POLYHEDRA

2006 ◽  
Vol 17 (03) ◽  
pp. 351-374
Author(s):  
M. A. APRODU ◽  
T. BOUZIANE
Keyword(s):  
Harmonic Functions ◽  
Holomorphic Functions ◽  
Precise Definition ◽  
Harmonic Morphisms ◽  
Harmonic Morphism ◽  
Target Manifold

The aim of this paper is to extend the notion of pseudo harmonic morphism introduced by Loubeau in [15] (see also [7, 4]) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic, as in [9], and pseudo-horizontally weakly conformal, see Sec. 3. We characterize them by means of germs of harmonic functions on the source polyhedron (see [13] for a precise definition) and germs of holomorphic functions on the Kähler target manifold, similarly to [15, 7].

On the Radial Symmetry Property for Harmonic Functions

2020 ◽  
Vol 64 (10) ◽  
pp. 9-19
Author(s):  
V. V. Volchkov ◽  
Vit. V. Volchkov
Keyword(s):  
Harmonic Functions ◽  
Symmetry Property ◽  
Radial Symmetry

Geometric Cardinal Invariants, Maximal Functions and a Measure Theoretic Pigeonhole Principle

2005 ◽  
Vol 11 (4) ◽  
pp. 517-525
Author(s):  
Juris Steprāns
Keyword(s):  
Harmonic Analysis ◽  
Set Theory ◽  
Harmonic Functions ◽  
Maximal Functions ◽  
Pigeonhole Principle ◽  
Cardinal Invariants ◽  
Geometric Objects

AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.

scholarly journals On the existence of various bounded harmonic functions with given periods, II

1975 ◽  
Vol 56 ◽  
pp. 1-5
Author(s):  
Masaru Hara
Keyword(s):  
Riemann Surface ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Dirichlet Integral ◽  
Period Function ◽  
Boundedness Property ◽  
One Dimensional ◽  
Bounded Harmonic Functions

Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.

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