Klainerman introduced in  the hyperboloidal method to prove global existence results for nonlinear Klein–Gordon equations by using commuting vector fields. In this paper, we extend the hyperboloidal method from Minkowski space to Lorentzian spacetimes. This approach is developed in  for proving, under the maximal foliation gauge, the global nonlinear stability of Minkowski space for Einstein equations with massive scalar fields, which states that, the sufficiently small data in a compact domain, surrounded by a Schwarzschild metric, leads to a unique, globally hyperbolic, smooth and geodesically complete solution to the Einstein Klein–Gordon system.
In this paper, we set up the geometric framework of the intrinsic hyperboloid approach in the curved spacetime. By performing a thorough geometric comparison between the radial normal vector field induced by the intrinsic hyperboloids and the canonical $\partial_r$, we manage to control the hyperboloids when they are close to their asymptote, which is a light cone in the Schwarzschild zone. By using such geometric information, we not only obtain the crucial boundary information for running the energy method in , but also prove that the intrinsic geometric quantities including the Hawking mass all converge to their Schwarzschild values when approaching the asymptote.
"An intrinsic hyperboloid approach for Einstein Klein–Gordon equations." J. Differential Geom. 115 (1) 27 - 109, May 2020. https://doi.org/10.4310/jdg/1586224841