An intrinsic hyperboloid approach for Einstein Klein-Gordon equations. (English) Zbl 1444.53043

In [Commun. Pure Appl. Math. 38, 631–641 (1985; Zbl 0597.35100)], S. Klainerman introduced the hyperboloidal method to obtain a global existence result for nonlinear Klein-Gordon equations by using commuting vector fields. The same result was independently obtained by J. Shatah [Commun. Pure Appl. Math. 38, 685–696 (1985; Zbl 0597.35101)], but using a different method. In this interesting paper, the author extends the hyperboloidal method from the Minkowski space to Lorentzian spacetimes. After setting up the analytic framework of the foliation of intrinsic hyperboloids and providing the geometric construction of the intrinsic frame of the Lorentz boosts, the author sketches the energy scheme in the proof of global stability of the Minkowski space for the Einstein Klein-Gordon system. Then, by assuming the foliation of the intrinsic hyperboloids and the maximal foliation exist till the last slice of hyperboloid, the main estimates are derived. These estimates depend only on the local-in-time energy estimates and the smallness of the given data on the initial maximal slice. The paper ends with some applications of these estimates. In particular, the asymptotic behavior of the Hawking mass along all hyperboloids is given.


53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
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