Mass formulae for asymptotically hyperbolic manifolds. (English) Zbl 1082.53034
Biquard, Olivier (ed.), AdS/CFT correspondence: Einstein metrics and their conformal boundaries. 73rd meeting of the theoretical physicists and mathematicians, Strasbourg, France, September 11–13, 2003. Zürich: European Mathematical Society (EMS) (ISBN 3-03719-013-2/pbk). IRMA Lectures in Mathematics and Theoretical Physics 8, 103-121 (2005).
Summary: The main goal of this text is to explain how both the concept of mass and the subsequent positivity and rigidity theorem can be generalized to the setting where the reference metric is no longer Euclidean but hyperbolic. It is based on the various works done on the subject due to L. Andersson and M. Dahl [Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16, 1–27 (1998; Zbl 0946.53021)], X. Wang [Mass for asymptotically hyperbolic manifolds, J. Differential Geom. 57, 273–229 (2001; Zbl 1037.53017)], X. Zhang [A definition of total energy-momenta and the positive mass theorem on asymptotically hyperbolic 3-manifolds, I. Commun. Math. Phys. 249, 529–548 (2004; Zbl 1073.83019)], P. Chruściel and G. Nagy [The mass of asymptotically andi-de Sitter space-times, Adv. Theor. Math. Phys. 5, 697–754 (2001; Zbl 1033.53061)] and P. T. Chruściel and the author [The mass of asymptotically hyperbolic Riemannian manifolds, Pac. J. Math. 212, 231–264 (2003; Zbl 1056.53025)].
For the entire collection see [Zbl 1062.81002].
For the entire collection see [Zbl 1062.81002].
MSC:
| 53C20 | Global Riemannian geometry, including pinching |
| 81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |
| 81T20 | Quantum field theory on curved space or space-time backgrounds |
| 83C47 | Methods of quantum field theory in general relativity and gravitational theory |
| 53C21 | Methods of global Riemannian geometry, including PDE methods; curvature restrictions |
