Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 083, 28 pages      arXiv:2303.15752
Contribution to the Special Issue on Differential Geometry Inspired by Mathematical Physics in honor of Jean-Pierre Bourguignon for his 75th birthday

Rigidity and Non-Rigidity of $\mathbb{H}^n/\mathbb{Z}^{n-2}$ with Scalar Curvature Bounded from Below

Tianze Hao a, Yuhao Hu ab, Peng Liu a and Yuguang Shi a
a) Key Laboratory of Pure and Applied Mathematics,School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China
b) School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, 200240, P.R. China

Received April 08, 2023, in final form October 20, 2023; Published online November 01, 2023

We show that the hyperbolic manifold $\mathbb{H}^n/\mathbb{Z}^{n-2}$ is not rigid under all compactly supported deformations that preserve the scalar curvature lower bound $-n(n-1)$, and that it is rigid under deformations that are further constrained by certain topological conditions. In addition, we prove two related splitting results.

Key words: scalar curvature; rigidity; ALH manifolds; $\mu$-bubbles.

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