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 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.

scholarly journals Some remarks concerning harmonic functions on homogeneous graphs

2003 ◽  
Vol DMTCS Proceedings vol. AC,... (Proceedings) ◽  
Author(s):  
Anders Karlsson
Keyword(s):  
Dirichlet Problem ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Cayley Graphs ◽  
Group Cohomology ◽  
Radial Variation ◽  
International Audience

International audience We obtain a new result concerning harmonic functions on infinite Cayley graphs $X$: either every nonconstant harmonic function has infinite radial variation in a certain uniform sense, or there is a nontrivial boundary with hyperbolic properties at infinity of $X$. In the latter case, relying on a theorem of Woess, it follows that the Dirichlet problem is solvable with respect to this boundary. Certain relations to group cohomology are also discussed.

scholarly journals Solutions of a Certain Nonlinear Elliptic Equation on Riemannian Manifolds

2001 ◽  
Vol 162 ◽  
pp. 149-167
Author(s):  
Yong Hah Lee
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Nonlinear Elliptic Equation ◽  
Complete Riemannian Manifold ◽  
Doubling Condition ◽  
Finite Covering ◽  
Nonlinear Elliptic ◽  
Finite Dimensional ◽  
Volume Doubling

In this paper, we prove that if a complete Riemannian manifold M has finitely many ends, each of which is a Harnack end, then the set of all energy finite bounded A-harmonic functions on M is one to one corresponding to Rl, where A is a nonlinear elliptic operator of type p on M and l is the number of p-nonparabolic ends of M. We also prove that if a complete Riemannian manifold M is roughly isometric to a complete Riemannian manifold with finitely many ends, each of which satisfies the volume doubling condition, the Poincaré inequality and the finite covering condition near infinity, then the set of all energy finite bounded A-harmonic functions on M is finite dimensional. This result generalizes those of Yau, of Donnelly, of Grigor’yan, of Li and Tam, of Holopainen, and of Kim and the present author, but with a barrier argument at infinity that the peculiarity of nonlinearity demands.

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

A note on positive harmonic functions

1948 ◽  
Vol 44 (2) ◽  
pp. 289-291 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Duke Math ◽  
Negative Number ◽  
Image Position ◽  
American Math ◽  
Positive Harmonic Functions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

scholarly journals Optimal growth of harmonic functions frequently hypercyclic for the partial differentiation operator

2019 ◽  
Vol 149 (6) ◽  
pp. 1577-1594
Author(s):  
Clifford Gilmore ◽  
Eero Saksman ◽  
Hans-Olav Tylli
Keyword(s):  
Growth Rate ◽  
Harmonic Function ◽  
Harmonic Functions ◽  
Optimal Growth ◽  
Differentiation Operator ◽  
Minimal Growth ◽  
Partial Differentiation

AbstractWe solve a problem posed by Blasco, Bonilla and Grosse-Erdmann in 2010 by constructing a harmonic function on ℝN, that is frequently hypercyclic with respect to the partial differentiation operator ∂/∂xk and which has a minimal growth rate in terms of the average L2-norm on spheres of radius r > 0 as r → ∞.

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