无穷远球面上加权爱因斯坦约束的调和、径向与壳稳定性
Harmonic, radial, and shell stability of the weighted Einstein constraints on the sphere at infinity
AI总结：
该研究针对无穷远球面的加权爱因斯坦约束，证明调和、径向与壳稳定性可由加权不等式导出，完成了引力屏蔽相关研究计划，引入了具有特定性质的壳泛函。
AI中文摘要：
我们考虑定义在任意维数球面上的四阶和二阶偏微分算子。这些算子是线性化爱因斯坦约束算子与其伴随算子的加权组合，在我们解决广义相对论中的最优局域化问题（也称为引力屏蔽问题）中发挥了关键作用。为了控制爱因斯坦约束方程解的渐近行为，在我们的配套论文（预印本arXiv:2312.17706）中，我们引入了调和稳定性、径向稳定性和壳稳定性的概念：调和稳定性控制临界调和模，径向稳定性支配球面平均的径向演化，壳稳定性控制解的径向-角向耦合演化。在本文中，我们证明这些稳定性性质可由加权庞加莱不等式、科恩不等式和哈代不等式导出。此外，我们研究了任意维数下相关几何常数的行为，得出稳定性条件对一大类局域化函数成立的结论：该理论适用于任意小的局域化区域，对应任意小口径的粘合锥；在另一极端情况，这些条件在整个球面上也成立，对应无局域化的情形。这在任意维数下，针对任意小口径粘合锥，完成了A. Carlotto和R. Schoen启动的关于引力屏蔽及具有超调和衰减估计的解的构造的研究计划。在证明中，我们引入了哈密顿量和动量泛函，将其称为壳泛函，并证明它们具有单调性和半强制性质；其结构还暗示了与其他曲率相关几何问题中出现的泛函可能存在类比关系。
英文摘要：
We consider fourth-order and second-order partial differential operators localized on domains of the sphere in arbitrary dimension. These operators arise as weighted compositions of the linearized Einstein constraint operators and their adjoints, and played a key role in our resolution of the optimal localization problem in general relativity, also referred to as the gravitational shielding problem. To control the asymptotic behavior of solutions to Einstein's constraints in our companion paper (preprint arXiv:2312.17706), we introduced the notions of harmonic, radial, and shell stability. Harmonic stability controls the borderline harmonic modes, radial stability governs the radial evolution of spherical averages, and shell stability controls the coupled radial-angular evolution of solutions. In the present paper, we establish that these stability properties follow from weighted Poincaré, Korn, and Hardy inequalities. Furthermore, in arbitrary dimension, we investigate the behavior of the associated geometric constants, and conclude that the stability conditions hold for a broad class of localization functions; the theory applies to arbitrarily small localization domains, corresponding to gluing cones with arbitrarily small aperture. At the opposite extreme, our conditions also hold on the entire sphere, corresponding to the absence of localization. This completes, for gluing cones of arbitrarily small aperture in every dimension, the program initiated by A. Carlotto and R. Schoen on gravitational shielding and the construction of solutions enjoying super-harmonic decay estimates. In our proofs, we introduce Hamiltonian and momentum functionals, which we call shell functionals, and show that they enjoy monotonicity and semi-coercivity properties; their structure also suggests possible analogies with functionals arising in other curvature-related geometric problems.
