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

Xiaodong Wang

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

Differential Geometry 2007-11-30 v1

Authors: Xiaodong Wang

Abstract

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

Keywords

harmonic functions simplicial volume riemannian geometry

Cite

@article{arxiv.0711.4592,
  title  = {Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space},
  author = {Xiaodong Wang},
  journal= {arXiv preprint arXiv:0711.4592},
  year   = {2007}
}

Comments

to appear in J. Geom. Anal

Related papers

View all related →

Metric Geometry · Mathematics

Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry