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 [14] 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 $\p_r$, 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 [14], but also prove that the intrinsic geometric quantities including the Hawking mass all converge to their Schwarzschild values when approaching the asymptote.

中文翻译：

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爱因斯坦克莱因-戈登方程的内在双曲面方法

在 [7] 中，Klainerman 引入了双曲面方法来证明非线性 Klein-Gordon 方程的全局存在性结果，即使用交换向量场。在本文中，我们将双曲面方法从闵可夫斯基空间扩展到洛伦兹时空。这种方法是在 [14] 中开发的，用于证明在最大叶理规范下，具有大量标量场的爱因斯坦方程的 Minkowski 空间的全局非线性稳定性，其中指出，紧凑域中足够小的数据，被 Schwarzschild 度量包围，导致爱因斯坦克莱因 - 戈登系统的独特，全局双曲线，平滑和测地线完整的解决方案。在本文中，我们建立了弯曲时空内本征双曲面方法的几何框架。通过在由内在双曲面引起的径向法向矢量场和规范 $\p_r$ 之间进行彻底的几何比较，我们设法在双曲面接近其渐近线时控制它们，渐近线是 Schwarzschild 区域中的光锥。通过使用这些几何信息，我们不仅获得了运行 [14] 中的能量方法的关键边界信息，而且还证明了包括霍金质量在内的内在几何量在接近渐近线时都收敛到其 Schwarzschild 值。