Perturbative solutions to the extended constant scalar curvature equations on asymptotically hyperbolic manifolds. (English) Zbl 1179.53051
In this paper, the author considers the so called Extended Constant Scalar Curvature Equations (ECSCE) for metrics which are close to an asymptotically hyperbolic Einstein one. The ECSCE are a generalization of the constant scalar equation for which one is given a symmetric two tensor \( T \) and seeks a metric \(g\) and a one-form \( \xi \). The author shows that if \( (M,g_0) \) is an asymptotically hyperbolic Einstein manifold, satisfying a non degeneracy condition, there is a well defined smooth map \( T \mapsto (\xi,g-g_0) \) (between neighborhoods of zero in Hölder spaces) which solve the ECSCE. The non degeneracy condition is satisfied on the hyperbolic space and many asymptotically hyperbolic manifolds.
Reviewer: Jean-Marc Bouclet (Villeneuve d’Ascq)
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 35R01 | PDEs on manifolds |
| 35J50 | Variational methods for elliptic systems |
| 58J05 | Elliptic equations on manifolds, general theory |
| 35J70 | Degenerate elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 35Q75 | PDEs in connection with relativity and gravitational theory |
Keywords:
asymptotically hyperbolic manifolds; general relativity; constraint equations; symmetric 2-tensors; asymptotic behaviorReferences:
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