A lower bound for Hausdorff dimensions of harmonic measures on negatively curved manifolds
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.
scientific article; zbMATH DE number 4203199
scientific article