Quasiconvex Foliations and Asymptotically Flat Metrics of Non-negative Scalar Curvature
Quasiconvex Foliations and Asymptotically Flat Metrics of Non-negative Scalar Curvature
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拟凸叶状结构和非负标量曲率的渐近平坦度量
DOI：
10.4310/cag.2004.v12.n3.a2
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发表时间：
2004
影响因子：
0.7
通讯作者：
G. Weinstein
中科院分区：
文献类型：
--
作者：
Brian T. Smith;G. Weinstein
其中R是在最大时间片上诱导的黎曼度量g的标量曲率，k是该时间片在环境洛伦兹四维流形中的第二基本形式。从（1）中可以清楚地看出，R ≥ 0。剩下的爱因斯坦真空方程可以被看作是控制由第一和第二基本形式g和k组成的数据的演化。由于这种演化在具有适当拓扑的初始数据空间中跟踪连续路径，因此很自然地会问：这个数据空间的拓扑属性是什么？特别是，一个相当重要的问题是，这个空间是否是连通的。可以使用保角方法证明两组初始数据（g，k），（g′，k′）在初始数据空间的相同路径连通分量中，当且仅当度量g和g′在非负标量曲率度量空间的相同路径连通分量中[17]。黎曼流形中的拓扑2-球面称为拟凸的，如果它的高斯曲率和平均曲率都是正的。叶理是准凸的
where R is the scalar curvature of the Riemannian metric g induced on a maximal time-slice, and k is the second fundamental form of that slice in the ambient Lorentzian 4-manifold. It is clear from (1) that R ≥ 0. The remaining Einstein vacuum equations can be seen as governing the evolution of the data consisting of the first and second fundamental form g and k. Since this evolution traces a continuous path in the space of initial data with the appropriate topology, it is natural to ask: what are the topological properties of this space of data? In particular, a question of considerable importance is whether this space is connected. It is possible to show using the conformal method that two sets of initial data (g, k), (g′, k′) are in the same pathconnected component of the space of initial data if and only the metrics g and g′ are in the same path-connected component of the space of metrics of non-negative scalar curvature [17]. A topological 2-sphere in a Riemannian manifold is said to be quasiconvex if its Gauss and mean curvatures are positive. A foliation is quasiconvex