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
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由指定平均曲率的 2 曲面组成的渐近平坦 3 流形的叶状结构
DOI：
10.4310/jdg/1191860394
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发表时间：
2004
影响因子：
2.5
通讯作者：
J. Metzger
中科院分区：
文献类型：
--
作者：
J. Metzger
对于一个具有相当一般的衰减条件的孤立引力系统，我们在具有渐近平坦初值的3个流形上构造了具有规定平均曲率的2-曲面。所讨论的表面形成了这样一个流形的渐近区域的规则的叶状结构。我们从叶理的几何形状恢复物理上相关的数据，特别是ADM动量。对于给定的数据集(M，g，K)，具有三维流形M，其黎曼度量g，以及周围四维洛伦兹时空流形中的第二基本形式K，我们解的方程是H+P=Const或H−P=Const。这里H是平均曲率，P=trk是K沿溶液表面的2迹。这是一个关于曲面位置的退化椭圆方程。它以各向异性的方式规定平均曲率，因为P取决于法线的方向。
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 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.