A lower bound for Hausdorff dimensions of harmonic measures on negatively curved manifolds
type: Item
instance of: scholarly article
title: A lower bound for Hausdorff dimensions of harmonic measures on negatively curved manifolds
zbMATH Open document ID: 0728.53028
DOI: 10.1007/BF02773752
published in: Israel Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: Let M be a complete simply connected m-dimensional Riemannian manifold with curvature sandwiched between two negative constants. For fixed \(x\in M\) the hitting probability of Brownian motion emanating at x at the ideal boundary of M can be viewed as a Borel probability measure on the unit sphere in \(T_ xM\). The Hausdorff dimension of this measure is shown to be bounded from below by (m-1)-times the asymptotical pinching p(x,\(\xi\)) at an arbitrary point \(\xi\in \partial M\). Here p(x,\(\xi\)) is defined to be the supremum of the pinching of the curvature of M restricted to an arbitrary nontrivial truncated cone with vertex x and center \(\xi\). In particular this Hausdorff dimension as a function of the metric is continuous at a metric of constant curvature.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 60J65
zbMATH DE Number: 4203199
zbMATH Keywords: curvature pinching
zbMATH Keywords: Brownian motion
zbMATH Keywords: Borel probability measure
zbMATH Keywords: Hausdorff dimension
author: Yuri Kifer
MaRDI profile type: Publication
cites work: Q3344779
cites work: Q4067705
cites work: Théoremes de comparaison en géométrie Riemannienne
cites work: Heat Kernel Bounds on Hyperbolic Space and Kleinian Groups
cites work: Function theory on manifolds which possess a pole
cites work: Q3959169
cites work: Q3751848
cites work: Hausdorff Dimension of Harmonic Measures on Negatively Curved Manifolds
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description: scientific article; zbMATH DE number 4203199