We study the space of Killing fields on the four dimensional AdS spacetime AdS 3,1 . Two subsets S and O are identified: S (the spinor Killing fields) is constructed from imaginary Killing spinors, and O (the observer Killing fields) consists of all hypersurface orthogonal, future timelike unit Killing fields. When the cosmology constant vanishes, or in the Minkowski spacetime case, these two subsets have the same convex hull in the space of Killing fields. In presence of the cosmology constant, the convex hull of O is properly contained in that of S . This leads to two different notions of energy for an asymptotically AdS spacetime, the spinor energy and the observer energy. In [10], Chru\'sciel, Maerten and Tod proved the positivity of the spinor energy and derived important consequences among the related conserved quantities. We show that the positivity of the observer energy follows from the positivity of the spinor energy. A new notion called the "rest mass" of an asymptotically AdS spacetime is then defined by minimizing the observer energy, and is shown to be evaluated in terms of the adjoint representation of the Lie algebra of Killing fields. It is proved that the rest mass has the desirable rigidity property that characterizes the AdS spacetime.

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Finding the stationary states of a free energy functional is an important problem in phase field crystal (PFC) models. Many efforts have been devoted to designing numerical schemes with energy dissipation and mass conservation properties. However, most existing approaches are time-consuming due to the requirement of small effective step sizes. In this paper, we discretize the energy functional and propose efficient numerical algorithms for solving the constrained nonconvex minimization problem. A class of gradient-based approaches, which are the so-called adaptive accelerated Bregman proximal gradient (AA-BPG) methods, is proposed, and the convergence property is established without the global Lipschitz constant requirements. A practical Newton method is also designed to further accelerate the local convergence with convergence guarantee. One key feature of our algorithms is that the energy dissipation and mass conservation properties hold during the iteration process. Moreover, we develop a hybrid acceleration framework to accelerate the AA-BPG methods and most of the existing approaches through coupling with the practical Newton method. Extensive numerical experiments, including two three-dimensional periodic crystals in the Landau--Brazovskii (LB) model and a two-dimensional quasicrystal in the Lifshitz--Petrich (LP) model, demonstrate that our approaches have adaptive step sizes which lead to a significant acceleration over many existing methods when computing complex structures.