A lower bound for Hausdorff dimensions of harmonic measures on negatively curved manifolds

1990 ◽  
Vol 71 (3) ◽  
pp. 339-348
Author(s):  
Yuri Kifer
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimensions ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Harmonic Measures

scholarly journals Harmonic measures for compact negatively curved manifolds

1997 ◽  
Vol 178 (1) ◽  
pp. 39-107 ◽  
Author(s):  
Ursula Hamenstädt
Keyword(s):  
Curved Manifolds ◽  
Negatively Curved ◽  
Harmonic Measures

scholarly journals Harmonic measures on negatively curved manifolds

2020 ◽  
Vol 69 (7) ◽  
pp. 2951-2971
Author(s):  
Yves Benoist ◽  
Dominique Hulin
Keyword(s):  
Curved Manifolds ◽  
Negatively Curved ◽  
Harmonic Measures

Torus actions, maximality, and non-negative curvature

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Christine Escher ◽  
Catherine Searle
Keyword(s):  
Negative Curvature ◽  
Torus Action ◽  
Maximal Symmetry ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Product Of Spheres ◽  
Rank Conjecture ◽  
Simply Connected ◽  
Symmetry Rank ◽  
Special Case

Abstract Let ℳ 0 n {\mathcal{M}_{0}^{n}} be the class of closed, simply connected, non-negatively curved Riemannian n-manifolds admitting an isometric, effective, isotropy-maximal torus action. We prove that if M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} , then M is equivariantly diffeomorphic to the free, linear quotient by a torus of a product of spheres of dimensions greater than or equal to 3. As a special case, we then prove the Maximal Symmetry Rank Conjecture for all M ∈ ℳ 0 n {M\in\mathcal{M}_{0}^{n}} . Finally, we show the Maximal Symmetry Rank Conjecture for simply connected, non-negatively curved manifolds holds for dimensions less than or equal to 9 without additional assumptions on the torus action.

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.

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