2004/10/19 by Jan Metzger, Metzger, Jan · 1 citation
Mathematics · Physics and Astronomy · #35J60 #53C42 #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds #gr-qc #math.DG #msc:35J60 #msc:53C42
paper · pdf · doi:10.48550/arxiv.math/0410413
34 Pages. No figures. Corrected the proof of Proposition 3.8
openalex publication_date 2004/10/19 · arxiv created 2005/09/23 · arxiv updated 2009/12/01 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
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.