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

We show that the hyperbolic manifold ${\mathbb{H}}^n\mathrm{/}{\mathbb{Z}}^{n-2}$ 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.