Rigidity and non-rigidity of $\H^n/\Z^{n-2}$ with scalar curvature bounded from below

Abstract

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 ´npn ´1q, and that it is rigid under deformations that are further constrained by certain topological conditions. In addition, we prove two related splitting results. Contents 1. Introduction 1 2. Non-rigidity of H n {Z n´2 5 3. ALH manifolds, mass and deformations 10 4. Two rigidity results 17 5. Two splitting results of 'cuspidal-boundary' type 20 Appendix A. µ-bubbles 24 Appendix B. Topological lemmas 26 Acknowledgement 28 References 28