Gradient estimate for exponentially harmonic functions on complete Riemannian manifolds

2013 ◽  
Vol 143 (3-4) ◽  
pp. 483-489
Author(s):  
Jiaxian Wu ◽  
Qihua Ruan ◽  
Yi-Hu Yang
Keyword(s):  
Riemannian Manifolds ◽  
Harmonic Functions ◽  
Gradient Estimate ◽  
Complete Riemannian Manifolds

Rough isometry and energy-finite solutions for the Schrödinger operator on Riemannian manifolds

2003 ◽  
Vol 133 (4) ◽  
pp. 855-873 ◽  
Author(s):  
Seok Woo Kim ◽  
Yong Hah Lee
Keyword(s):  
Riemannian Manifolds ◽  
Schrödinger Operator ◽  
Sobolev Inequality ◽  
Harmonic Functions ◽  
Schrodinger Operator ◽  
Dirichlet Integral ◽  
Local Conditions ◽  
Complete Riemannian Manifolds ◽  
Bounded Harmonic Functions ◽  
Rough Isometry

In this paper, we prove that the dimension of the space of bounded energy-finite solutions for the Schrödinger operator is invariant under rough isometries between complete Riemannian manifolds satisfying the local volume condition, the local Poincaré inequality and the local Sobolev inequality. We also prove that the dimension of the space of bounded harmonic functions with finite Dirichlet integral is invariant under rough isometries between complete Riemannian manifolds satisfying the same local conditions. These results generalize those of Kanai, Grigor'yan, the second author, and Li and Tam.

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.

Sign in / Sign up

Export Citation Format

Share Document