Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature

Details Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature Foliations of asymptotically flat 3-manifolds by 2-surfaces of prescribed mean curvature

2004-10-19

arxiv.org/abs/math/0410413v2

Mathematics

Differential Geometry

34 Pages. No figures. Corrected the proof of Proposition 3.8

Scientific paper

We construct 2-surfaces of prescribed mean curvature in 3-manifolds carrying asymptotically flat initial data for an isolated gravitating sysqtem 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 = tr K 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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