ii iii iv PrefaceThe Hyperboloidal Foliation Method presented in this monograph is based on a p3`1q-foliation of Minkowski spacetime by hyperboloidal hypersurfaces. It allows us to establish global-in-time existence results for systems of nonlinear wave equations posed on a curved spacetime and to derive uniform energy bounds and optimal rates of decay in time. We are also able to encompass the wave equation and the Klein-Gordon equation in a unified framework and to establish a well-posedness theory for nonlinear wave-Klein-Gordon systems and a large class of nonlinear interactions.The hyperboidal foliation of Minkowski spactime we rely upon in this book has the advantage of being geometric in nature and, especially, invariant under Lorentz transformations. As stated, our theory applies to many systems arising in mathematical physics and involving a massive scalar field, such as the Dirac-Klein-Gordon system. As it provides uniform energy bounds and optimal rates of decay in time, our method appears to be very robust and should extend to even more general systems.We have built upon many earlier studies of nonlinear wave equations or Klein-Gordon equations, especially by Sergiu Klainerman, Demetri Christodoulou, Jalal Shatah, Alain Bachelot, and many others. The coupling of nonlinear wave-Klein-Gordon systems was first understood by Soichiro Katayama who succeeded to establish an existence theory of such systems.Importantly, in developing the Hyperboloidal Foliation Method, we were inspired by earlier work on the Einstein equations of general relativity by Helmut Friedrich, Vincent Moncrief, and Anil Zenginoglu.We are very grateful Soichiro Katayama for observations he made to the authors on a preliminary version of this monograph.Last but not least, the authors are very grateful to their respective families for their strong support.

Abstract. The Hyperboloidal Foliation Method (introduced by the authors in 2014) is extended here and applied to the Einstein equations of general relativity. Specifically, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields, while existing methods only apply to massless scalar fields. First of all, by analyzing the structure of the Einstein equations in wave coordinates, we exhibit a nonlinear wave-Klein-Gordon model defined on a curved background, which is the focus of the present paper. For this model, we prove here the existence of global-in-time solutions to the Cauchy problem, when the initial data have sufficiently small Sobolev norms. A major difficulty comes from the fact that the class of conformal Killing fields of Minkowski space is significantly reduced in presence of a massive scalar field, since the scaling vector field is not conformal Killing for the Klein-Gordon operator. Our method relies on the foliation (of the interior of the light cone) of Minkowski spacetime by hyperboloidal hypersurfaces and uses Lorentz-invariant energy norms. We introduce a frame of vector fields adapted to the hyperboloidal foliation and we establish several key properties: Sobolev and Hardy-type inequalities on hyperboloids, as well as sup-norm estimates which correspond to the sharp time decay for the wave and the Klein-Gordon equations. These estimates allow us to control interaction terms associated with the curved geometry and the massive field, by distinguishing between two levels of regularity and energy growth and by a successive use of our key estimates in order to close a bootstrap argument.

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Printed in SingaporeThe Global Nonlinear Stability of Minkowski Space for Self-Gravitating Massive Fields Downloaded from www.worldscientific.com by 54.213.91.117 on 05/10/18. For personal use only.The theory presented in this monograph establishes the first mathematically rigorous result on the global nonlinear stability of self-gravitating matter under small perturbations of an asymptotically flat, spacelike hypersurface of Minkowski spacetime. It allows one to exclude the existence of dynamically unstable, self-gravitating massive fields and, therefore, solves a long-standing open problem in General Relativity. By a significant extension of the Hyperboloidal Foliation Method they introduced in 2014, the authors establish global-in-time existence for the Einstein equations expressed as a coupled wave-Klein-Gordon system of partial differential equations. The metric and matter fields are sought for in Sobolev-type functional spaces, suitably defined from the translations and the boosts of Minkowski spacetime. PrefaceThe theory presented in this Monograph establishes the first mathematically rigorous result about the global nonlinear stability of self-gravitating matter under small perturbations. It allows us to exclude the existence of dynamically unstable, self-gravitating massive fields and, therefore, solves a long-standing open problem in General Relativity. We establish that Minkowski spacetime is nonlinearly stable in presence of a massive scalar field under suitable smallness conditions (for, otherwise, black holes might form). We formulate the initial value problem for the Einstein-massive scalar field equations, when the initial slice is a perturbation of an asymptotically flat, spacelike hypersurface in Minkowski space, and we prove that this perturbation disperses in future timelike directions so that the associated Cauchy development is future geodesically complete.Our method of proof which we refer to as the Hyperboloidal Foliation Method, extends the standard 'vector field method' developed for massless fields and, importantly, does not use the scaling vector field of Minkowski space. We construct a foliation (of the interior of a light cone) by spacelike and asymptotically hyperboloidal hypersurfaces and we rely on a decomposition of the Einstein equations expressed in wave gauge and in a semi-hyperboloidal frame, in a sense defined in this Monograph. We focus here on the problem of the evolution of a spatially compact matter field, and we consider initial data coinciding, in a neighborhood of spacelike infinity, with a spacelike slice of Schwarzschild spacetime. We express the Einstein equations as a system of coupled nonlinear wave-Klein-Gordon equations (with differential constraints) posed on a curved space (whose metric is one of the unknowns).The main challenge is to establish a global-in-time existence theory for coupled wave-Klein-Gordon systems in Sobolev-type spaces defined from the translations and the boosts of Minkowski spacetime, only. To this end, we rely on the following novel an...

In this paper and its successor, we make an application of the hyperboloidal foliation method in [Formula: see text] space-time dimension. After the establishment of some technical tools in this paper, we will prove further the global existence of small regular solution to a class of hyperbolic system composed by a wave equation and a Klein–Gordon equation with null couplings. Our method belongs to vector field method and, more precisely, is a combination of the normal form and the hyperboloidal foliation method.