Harmonic functions along Brownian balls and the Liouville property for solvable Lie groups

1993 ◽  
Vol 296 (1) ◽  
pp. 577-584
Author(s):  
Bernhard Leeb
Keyword(s):  
Lie Groups ◽  
Harmonic Functions ◽  
Liouville Property ◽  
Solvable Lie Groups

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

Analysis of a Distinguished Laplacian on Solvable Lie Groups

1993 ◽  
Vol 163 (1) ◽  
pp. 151-162 ◽  
Author(s):  
Saverio Giulini ◽  
Giancarlo Mauceri
Keyword(s):  
Lie Groups ◽  
Solvable Lie Groups

Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

2018 ◽  
Vol 18 (3) ◽  
pp. 337-344 ◽  
Author(s):  
Ju Tan ◽  
Shaoqiang Deng
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Algebras ◽  
Critical Points ◽  
Vector Fields ◽  
Energy Functional ◽  
Smooth Vector ◽  
Solvable Lie Groups ◽  
Invariant Vector ◽  
Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

Harmonic functions on ℝ-covered foliations

2009 ◽  
Vol 29 (4) ◽  
pp. 1141-1161
Author(s):  
S. FENLEY ◽  
R. FERES ◽  
K. PARWANI
Keyword(s):  
Harmonic Functions ◽  
Riemannian Metrics ◽  
Related Result ◽  
Continuous Functions ◽  
Property A ◽  
Liouville Property ◽  
Foliated Manifold ◽  
Curved Manifolds ◽  
Codimension One ◽  
Negatively Curved

AbstractLet (M,ℱ) be a compact codimension-one foliated manifold whose leaves are endowed with Riemannian metrics, and consider continuous functions on M that are harmonic along the leaves of ℱ. If every such function is constant on leaves, we say that (M,ℱ) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville property. A related result for ℝ-covered foliations is also established.

Invariant orderings in solvable Lie groups

1989 ◽  
Vol 30 (1) ◽  
pp. 44-53 ◽  
Author(s):  
V. M. Gichev
Keyword(s):  
Lie Groups ◽  
Solvable Lie Groups
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