AI 中文总结
本文针对满足真空爱因斯坦约束方程的初始渐近平直柯西面，在平均曲率适当小的条件下，解决了常规椭圆方法构造极大叶状结构的根本性困难，给出了极大叶状结构的构造方案。
AI 中文摘要
众所周知，爱因斯坦方程是关于洛伦兹度量的张量方程，因此选择合适的规范条件对求解它们至关重要。作为一种强有力的规范条件，极大叶状结构在广义相对论的两项开创性工作——闵可夫斯基时空的全局稳定性与有界L²曲率猜想中发挥了关键作用。然而，若初始超曲面不满足极大条件，求解爱因斯坦方程时，构造极大叶状结构的常规椭圆方法会遇到根本性困难。本文的目的是：在平均曲率满足适当小性条件下，针对满足真空爱因斯坦约束方程的任意初始渐近平直柯西面，提供一种构造其周围极大叶状结构的方法。
英文摘要
It is well known that the Einstein equations are tensor equations for a Lorentzian metric. Hence, choosing a suitable gauge condition is crucial for solving them. As a powerful gauge condition, maximal foliations have played a pivotal role in two groundbreaking works in general relativity: the global stability of Minkowski spacetime and the bounded L^2 curvature conjecture. Nevertheless, if the initial hypersurface fails to be maximal, the usual elliptic approach for constructing maximal foliations encounters fundamental difficulties when solving the Einstein equations. The purpose of the paper is to provide a construction of maximal foliations around any initial asymptotically flat Cauchy surface satisfying vacuum Einstein constraint equations, under a suitable smallness condition on the mean curvature.
Comments55 pages
