Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

Abstract

Li and Wang [J Differ Geom 58:501–534 (2001), J Differ Geom 62:143–162 (2002)] proved a splitting theorem for an n-dimensional Riemannian manifold with $$Ric \ge -(n-1)$$ and the bottom of spectrum $$\lambda _0(M)=\frac{(n-1)^2}{4}$$ . For an n-dimensional compact manifold M with $$Ric\ge -(n-1)$$ with the volume entropy $$h(M)=n-1$$ , Ledrappier and Wang (J Differ Geom 85:461–477, 2010) proved that the universal cover $$\widetilde{M}$$ is isometric to the hyperbolic space $$\mathbb {H}^n$$ . We will prove analogue theorems for Alexandrov spaces.