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

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Abstract

Let (M n,g) be a compact Riemannian manifold with Ric ≥−(n−1). It is well known that the bottom of spectrum λ 0 of its universal covering satisfies λ 0≤(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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Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Xiaodong Wang

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  1. Xiaodong Wang
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Correspondence to Xiaodong Wang.

Additional information

Communicated by Peter Li.

The author was partially supported by NSF Grant 0505645.

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Wang, X. Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space. J Geom Anal 18, 272–284 (2008). https://doi.org/10.1007/s12220-007-9001-z

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  • DOI: https://doi.org/10.1007/s12220-007-9001-z

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