Sobolev regularity of compactified 3-manifolds and the ADM Center of Mass

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 question 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 shall first establish regularity properties of $L^{q'}$-solutions associated to the operator $\Delta_{\hat{g}}:W^{k,p}(S_2\hat{M})\to W^{k-2,p}(S_2\hat{M})$, for $k=1,2$ and $1<p\leq q$, where $\hat{M}^n$ is a closed manifold, $\hat{g}\in W^{2,q}(\hat{M})$, with $q>\frac{n}{2}$ and $S_2\hat{M}$ denotes the bundle of symmetric $(0,2)$-tensor fields. Appealing to these results, we establish Sobolev regularity 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 will allow us to address a version of a conjecture posed by C. Cederbaum and A. Sakovich concerning the convergence of the COM of such manifolds.