Mass rigidity for hyperbolic manifolds. (English) Zbl 1440.81066
Summary: We prove the rigidity of the positive mass theorem for asymptotically hyperbolic manifolds. Namely, if the mass equality \(p_0=\sqrt{p_1^2+\cdots + p_n^2}\) holds, then the manifold is isometric to hyperbolic space. The result was previously proven for spin manifolds [L. Andersson and M. Dahl, Ann. Global Anal. Geom. 16, No. 1, 1–27 (1998; Zbl 0946.53021); P. Chruściel and M. Herzlich, Pac. J. Math. 212, No. 2, 231–264 (2004; Zbl 1056.53025); M. Min-Oo, Math. Ann. 285, No. 4, 527–539 (1989; Zbl 0686.53038); X. Wang, J. Differ. Geom. 57, No. 2, 273–299 (2001; Zbl 1037.53017)] or under special asymptotics [L. Andersson et al., Ann. Henri Poincaré 9, No. 1, 1–33 (2008; Zbl 1134.81037)].
MSC:
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 70H40 | Relativistic dynamics for problems in Hamiltonian and Lagrangian mechanics |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
| 53C20 | Global Riemannian geometry, including pinching |
| 53C27 | Spin and Spin\({}^c\) geometry |
| 57K32 | Hyperbolic 3-manifolds |
| 83C47 | Methods of quantum field theory in general relativity and gravitational theory |
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