# Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

## TL;DR

This paper extends rigidity results related to the bottom of the spectrum and volume entropy from Riemannian manifolds to Alexandrov spaces, showing similar splitting theorems under curvature and entropy conditions.

## Contribution

It proves analogue theorems for Alexandrov spaces, generalizing known results from Riemannian geometry to a broader class of metric spaces.

## Key findings

- Rigidity theorems for Alexandrov spaces with curvature bounds.
- Splitting results under spectral and entropy conditions.
- Extension of hyperbolic space characterization to Alexandrov spaces.

## Abstract

In \cite{LiWang2001complete1,LiWang2001complete2}, Li-Wang proved a splitting theorem for an n-dimensional Riemannian manifold with $Ric\geqslant -(n-1)$ and the bottom of spectrum $\lambda_0(M)=\frac{(n-1)^2}{4}$. For an n-dimensional compact manifold $M$ with $Ric\geqslant-(n-1)$ with the volume entropy $h(M)=n-1$, Ledrappier-Wang \cite{LeW2010volent} proved that the universal cover $\tilde{M}$ is isometric to the hyperbolic space $\mathbb{H}^n$. We will prove analogue theorems for Alexandrov spaces.