标题： 非线性散射理论用于所有偶数空间维数中渐近 de Sitter 真空解 显示英文标题

标题： Nonlinear Scattering Theory for Asymptotically de Sitter Vacuum Solutions in All Even Spatial Dimensions

摘要： 本文的目的是为爱因斯坦真空方程在$(n+1)$维空间中渐近德西特解建立一个确定性的定量非线性散射理论，其中$n\geq4$为偶数，这些解由时空类空渐近边界$\mathscr{I}^-$和$\mathscr{I}^+.$处的小散射数据确定。与奇数情形相比，空间维数为$n$的情况面临重大挑战，之前的文献研究中尚未解决这一问题。 此处所指的散射理论意味着散射态的存在性和唯一性、渐近完备性以及存在一个可逆的散射映射，并对其范数有定量控制。 散射态的存在性和唯一性意味着对于任何小的渐近数据，都存在唯一的全局解满足爱因斯坦方程，并且该解始终保持接近德西特度规。 渐近完备性是相反的陈述，表明任何这样的解都在$\mathscr{I}^-$和$\mathscr{I}^+.$处诱导渐近数据。对于足够小的渐近数据，我们构造了散射映射$\mathscr{S}$，将$\mathscr{I}^-$处的数据映射到$\mathscr{I}^+,$处的数据，并且我们证明了在关于 Sobolev 类范数下，映射$\mathscr{S}$在 de Sitter 数据处局部可逆且局部 Lipschitz。 散射图结果是精确的，并且避免了任何“导数损失”，即我们在$\mathscr{I}^-$和$\mathscr{I}^+$处使用相同的 Sobolev 范数来衡量渐近数据的小性。 精确结果的证明需要对爱因斯坦方程进行详细分析，涉及解的几何 Littlewood-Paley 分解，这在我们的相关论文中已经完成。 显示英文摘要

摘要： The purpose of this paper is to establish a definitive quantitative nonlinear scattering theory for asymptotically de Sitter solutions of the Einstein vacuum equations in $(n+1)$ dimensions with $n\geq4$ even, which are determined by small scattering data at the spacelike asymptotic boundaries $\mathscr{I}^-$ and $\mathscr{I}^+.$ The case of even spatial dimension $n$ poses significant challenges compared to its odd counterpart and was left open by the previous works in the literature. Here, scattering theory is understood to mean existence and uniqueness of scattering states, asymptotic completeness, and the existence of an invertible scattering map with quantitative control on its norm. The existence and uniqueness of scattering states imply that for any small asymptotic data there exists a unique global solution to the Einstein equations, which remains close to the de Sitter metric. Asymptotic completeness is the converse statement, showing that any such solution induces asymptotic data at $\mathscr{I}^-$ and at $\mathscr{I}^+.$ For sufficiently small asymptotic data, we construct the scattering map $\mathscr{S}$ taking data at $\mathscr{I}^-$ to data at $\mathscr{I}^+,$ and we show that the map $\mathscr{S}$ is locally invertible and locally Lipschitz at the de Sitter data, with respect to a Sobolev-type norm. The scattering map result is sharp and avoids any "derivative loss", in the sense that we measure the smallness of asymptotic data at $\mathscr{I}^-$ and $\mathscr{I}^+$ using the same Sobolev norm. The proof of the sharp result requires a detailed analysis of the Einstein equations involving a geometric Littlewood-Paley decomposition of the solution, carried out in our companion paper.