Rigidity and Non-Rigidity of Hⁿ/Z^n-2 with Scalar Curvature Bounded from Below

We show that the hyperbolic manifold H-n/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.