Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

Jiang Yin

Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

Metric Geometry 2017-02-21 v2 Differential Geometry

Authors: Jiang Yin

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.

Keywords

Alexandrov spaces Riemannian manifold Ricci curvature lower bound bottom of spectrum volume entropy splitting theorem hyperbolic space

Cite

@article{arxiv.1702.04461,
  title  = {Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry},
  author = {Jiang Yin},
  journal= {arXiv preprint arXiv:1702.04461},
  year   = {2017}
}

Comments

27 pages, there is an error in the previous version, Bochner formula, i.e. Theorem 2.16 and Lemma 2.17, corrected in this version

Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

Abstract

Keywords

Cite

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