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Related aspects of positivity in Riemannian geometry. (English) Zbl 0615.53029
In this paper, the author investigates connections among various numerical functions which can be related in the geometric context of Riemannian manifold theory. These functions include (a) the Hausdorff dimension of a closed set in Euclidean space, (b) the point of the \(L^ 2\)-spectrum nearest to zero for the Friedrich extension of a semidefinite symmetric operator \(\Delta\), (c) the critical exponent of the Poincaré- Dirichlet series of a discrete group of Moebius transformations of \(S^ d\), (d) the parameter of the spherical complementary series of irreducible representations of 0(n,1), (e) \(\lambda\)-potential theory and (f) the entropy of an ergodic measure preserving flow. General connections between (b) and (e) for Riemannian manifolds are obtained. For the case of constant negative curvature, (a), (c), (d) are related to (b) and (e).
Reviewer: A.Stone

MSC:
53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
58J65 Diffusion processes and stochastic analysis on manifolds
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