scholarly journals Denjoy–Ahlfors theorem for harmonic functions on Riemannian manifolds and external structure of minimal surfaces

1996 ◽  
Vol 4 (4) ◽  
pp. 547-587 ◽  
Author(s):  
Vladimir Miklyukov ◽  
Vladimir Tkachev
Keyword(s):  
Riemannian Manifolds ◽  
Minimal Surfaces ◽  
Harmonic Functions ◽  
External Structure

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.

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 r-Harmonic and Complex Isoparametric Functions on the Lie Groups $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$

2020 ◽  
Vol 58 (4) ◽  
pp. 477-496
Author(s):  
Sigmundur Gudmundsson ◽  
Marko Sobak
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Lie Group ◽  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Semidirect Products ◽  
Simply Connected ◽  
Isoparametric Functions ◽  
General Method

Abstract In this paper we introduce the notion of complex isoparametric functions on Riemannian manifolds. These are then employed to devise a general method for constructing proper r-harmonic functions. We then apply this to construct the first known explicit proper r-harmonic functions on the Lie group semidirect products $${{\mathbb {R}}}^m \ltimes {{\mathbb {R}}}^n$$ R m ⋉ R n and $${{\mathbb {R}}}^m \ltimes \mathrm {H}^{2n+1}$$ R m ⋉ H 2 n + 1 , where $$\mathrm {H}^{2n+1}$$ H 2 n + 1 denotes the classical $$(2n+1)$$ ( 2 n + 1 ) -dimensional Heisenberg group. In particular, we construct such examples on all the simply connected irreducible four-dimensional Lie groups.

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