An intrinsic hyperboloid approach for Einstein Klein-Gordon equations

Author: 

Wang, Q

Publication Date: 

7 April 2020

Journal: 

Journal of Differential Geometry

Last Updated: 

2020-08-10T12:58:22.77+01:00

Issue: 

1

Volume: 

115

DOI: 

10.4310/jdg/1586224841

page: 

27-109

abstract: 

<p>In [7] Klainerman introduced the hyperboloidal method to prove the 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 [15] 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 KleinGordon system.</p> <p>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 ∂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 [15], but also prove that the intrinsic geometric quantities including the Hawking mass all converge to their Schwarzschild values when approaching the asymptote.</p>

Symplectic id: 

891701

Submitted to ORA: 

Submitted

Publication Type: 

Journal Article