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We construct 2-surfaces of prescribed mean curvature in 3-manifolds 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 manifold $M$, 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.
Continuing the program of S. Donaldson and I. Smith, and M. Usher, we introduce refinements of the Donaldson-Smith standard surface count which are designed to count nodal pseudoholomorphic curves and curves with a prescribed decomposition into reducible components. In cases where a corresponding analogue of the Gromov-Taubes invariant is easy to define, our invariants agree with those analogues. We also prove a vanishing result for some of the invariants that count nodal curves.