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

Jan Metzger Am M¨uhlenberg 1

Differential Geometry mathscidoc:1609.10039

Journal of Differential Geometry, 77, (2), 201-236, 2007
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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@inproceedings{jan2007foliations,
  title={FOLIATIONS OF ASYMPTOTICALLY FLAT 3-MANIFOLDS BY 2-SURFACES OF PRESCRIBED MEAN CURVATURE},
  author={Jan Metzger},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909100945370830696},
  booktitle={Journal of Differential Geometry},
  volume={77},
  number={2},
  pages={201-236},
  year={2007},
}
Jan Metzger. FOLIATIONS OF ASYMPTOTICALLY FLAT 3-MANIFOLDS BY 2-SURFACES OF PRESCRIBED MEAN CURVATURE. 2007. Vol. 77. In Journal of Differential Geometry. pp.201-236. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160909100945370830696.
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