  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.
