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Mathematische Zeitschrift

, Volume 291, Issue 1–2, pp 55–84 | Cite as

Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

  • Yin JiangEmail author
Article
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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.

Notes

Acknowledgements

We are grateful to Xiantao Huang, RenJin Jiang, Shicheng Xu and Huichun Zhang for helpful discussions. We also thank Xiaochun Rong for helpful suggestions. The author is partially supported by a fund of China Postdoctoral Science Foundation, no. 2017M620828.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics DepartmentCapital Normal UniversityBeijingPeople’s Republic of China

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