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arXiv · 0711.4592

Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space

Xiaodong Wang
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Abstract

Let $(M^{n},g)$ be a compact Riemannian manifold with $Ric\geq-(n-1) $. It is well known that the bottom of spectrum $λ_{0}$ of its unverversal covering satisfies $λ_{0}\leq(n-1) ^{2}/4 $. We prove that equality holds iff $M$ is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.

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Xiaodong Wang. 2007-11-28. Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space. https://arxiv.org/abs/0711.4592

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