Entropy Rigidity of negatively curved manifolds of finite volume

2017/02/08 by Peigne, M., Sambusetti, A.
#53C20 #Differential Geometry (math.DG) #FOS: Mathematics #G.0

paper · doi:10.48550/arxiv.1702.06567

Abstract

We prove the following entropy-rigidity result in finite volume: if X is a negatively curved manifold with curvature -b²≤ K_X ≤ -1, then Ent_top(X) = n-1 if and only if X is hyperbolic. In particular, if X has the same length spectrum of a hyperbolic manifold X₀, the it is isometric to X₀ (we also give a direct, entropy-free proof of this fact). We compare with the classical theorems holding in the compact case, pointing out the main difficulties to extend them to finite volume manifolds.

Related

• Entropies for negatively curved manifolds
• Entropy rigidity for finite volume strictly convex projective manifolds
• Entropy Rigidity and Hilbert Volume
• The Cayley hyperbolic space and volume entropy rigidity
• Quantitative rigidity of almost maximal volume entropy for both RCD spaces and integral Ricci curvature bound