We construct 2-surfaces of prescribed mean curvature in 3manifolds carrying asymptotically flat initial data for an isolated gravitating system with rather general decay conditions. The surfaces in question form a regular foliation of the asymptotic region of such a manifold. We recover physically relevant data, especially the ADM-momentum, from the geometry of the foliation. For a given set of data (M, g,K), with a three dimensional manifoldM, its Riemannian metric g, and the second fundamental form K in the surrounding four dimensional Lorentz space time manifold, the equation we solve is H+P = const or H−P = const. Here H is the mean curvature, and P = trK is the 2-trace of K along the solution surface. This is a degenerate elliptic equation for the position of the surface. It prescribes the mean curvature anisotropically, since P depends on the direction of the normal.

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We give a complete diffeomorphism classification of 1-connected closed manifolds M with integral homology H¤(M) ∼= Z ⊕ Z ⊕ Z, provided that dim(M) 6= 4.

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A generation of symbols asserted for n ≥ 0 in the proof of Theorem 3.3 of the original paper in fact only holds for n > 0, thus undermining the proof of the theorem. A new version of Section 3.5 of the original paper is given, culminating in a corrected proof of Theorem 3.3. The author thanks Deepak Khosla for pointing out the gap in the previous version of the proof.