Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

2017/02/15 by Yin, Jiang
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1702.04461

Abstract

In \citeLiWang2001complete1,LiWang2001complete2, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with Ric\geqslant -(n-1) and the bottom of spectrum λ₀(M)=((n-1)²)/(4). For an n-dimensional compact manifold M with Ric\geqslant-(n-1) with the volume entropy h(M)=n-1, Ledrappier-Wang \citeLeW2010volent proved that the universal cover M is isometric to the hyperbolic space ℍⁿ. We will prove analogue theorems for Alexandrov spaces.

Related

• Quantitative maximal volume entropy rigidity on Alexandrov spaces
• Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space
• Relatively maximum volume rigidity in Alexandrov geometry
• Spectral rigidity of manifolds with Ricci bounded below and maximal bottom spectrum
• Sharp bounds and rigidity for volumes of boundaries of Alexandrov spaces