紧致化3-流形的Sobolev正则性与ADM质心
Sobolev regularity of compactified 3-manifolds and the ADM Center of Mass
AI总结：
本文通过Cotton张量衰减建立共形紧致化AE 3-流形的Sobolev正则性，构造优选渐近坐标以获得一阶史瓦西展开，从而证明ADM质心收敛的几何判据，部分解决Cederbaum-Sakovich猜想。
摘要：
本文研究渐近欧几里得（AE）流形$(M^3,g)$上优选渐近坐标的存在性，使得$g$允许渐近史瓦西型一阶展开，该存在性仅基于先验几何条件，并用于建立保证ADM质心（COM）收敛的几何判据。此问题通过关联到此类流形共形紧致化的正则性研究来分析，而后者本身借助系数正则性非常有限的算子的椭圆理论进行探索。鉴于这些相关问题，我们首先通过Cotton张量的衰减建立共形紧致化AE 3-流形的Sobolev正则性性质，改进了先前结果。这使我们能在此类AE流形上构造优选渐近坐标，其中度量具有一阶史瓦西型展开，进而使我们能处理C. Cederbaum和A. Sakovich提出的关于此类流形质心收敛性的一个猜想版本。
英文摘要：
In this paper we address the existence of preferred asymptotic coordinates on asymptotically Euclidean (AE) manifolds $(M^3,g)$ such that $g$ admits an asymptotically Schwarzschildian first order expansion, based purely on a priori geometric conditions, which will then be used to establish geometric criteria guaranteeing the convergence of the ADM center of mass (COM). This problem is analysed relating it to the study of the regularity of conformal compactifications of such manifolds, which is itself explored via elliptic theory for operators with coefficients of very limited regularity. With these related problems in mind, we first establish Sobolev regularity properties of conformally compactified AE 3-manifolds via the decay of the Cotton tensor, improving on previous results. This allows us to construct preferred asymptotic coordinates on such AE manifolds where the metric has a first order Schwarzschildian expansion, which in turn allows us to address a version of a conjecture posed by C. Cederbaum and A. Sakovich concerning the convergence of the COM of such manifolds.
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