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

2007/11/28 by Xiaodong Wang, Wang, Xiaodong · 2 citations
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.0711.4592

openalex publication_date 2007/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let (Mⁿ,g) be a compact Riemannian manifold with Ric≥-(n-1) . It is well known that the bottom of spectrum λ₀ of its unverversal covering satisfies λ₀≤(n-1) ²/4 . We prove that equality holds iff M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy.

Cited by

• Geometric Rigidity via Almost-Harmonic Twisted Spinors
• Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum

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