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Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry

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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.

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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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  1. Mathematics Department, Capital Normal University, Beijing, People’s Republic of China

    Yin Jiang

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Jiang, Y. Maximal bottom of spectrum or volume entropy rigidity in Alexandrov geometry. Math. Z. 291, 55–84 (2019). https://doi.org/10.1007/s00209-018-2073-6

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  • DOI: https://doi.org/10.1007/s00209-018-2073-6

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