The Dirac-Witten Operator on Spacelike Hypersurfaces
The Dirac-Witten Operator on Spacelike Hypersurfaces
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DOI：
10.4310/cag.2003.v11.n4.a5
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发表时间：
2003
影响因子：
0.7
通讯作者：
Oussama Hijazi;X. Zhang
中科院分区：
文献类型：
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作者：
Oussama Hijazi;X. Zhang
由E. Witten [W]给出的著名的正质量定理的旋量证明是基于对超曲面狄拉克型算子（称为狄拉克-威腾算子）的Weitzenböck型公式的微妙使用。质量是由边界积分项在无穷远处的极限给出的。在过去的二十年中，对封闭黎曼自旋流形上经典狄拉克算子的特征值的下界进行了深入的研究（例如，参见[F， Hi1]）。具有最小特征值的流形的特征是一些超定系统的解的存在性。最近[HMZ]，在一些自然边界条件下，这些结果被推广到有边界的黎曼紧化自旋流形的情况。此外，在[Z3， HZ]（参见[M]）中，研究了当环境流形为黎曼和自旋时Dirac-Witten算子的相应问题。本文考虑（n + 1）维洛伦兹流形的紧致（有边界或无边界）类空间超曲面，其度规
The well-known spinorial proof of the positive mass theorem given by E. Witten [W] is based on a subtle use of the Weitzenböck type formula for the hypersurface Dirac-type operator called the Dirac-Witten operator. The mass is given by the limit at infinity of some boundary integral term. In the past two decades, lower bounds for the eigenvalues of the classical Dirac operator on closed Riemannian spin manifolds were intensively studied (see for example, [F, Hi1]). Manifolds with minimal eigenvalues are characterized by the existence of solutions of some overdetermined systems as for Killing spinors. More recently [HMZ], under some natural boundary conditions, such results were extended to the case of Riemannian compact spin manifolds with boundary. Moreover, in [Z3, HZ] (see also [M]) examined the corresponding questions to the Dirac-Witten operator when the ambient manifold is Riemmanian and spin. In this paper, we consider compact (with or without boundary) spacelike hypersurfaces of an (n + 1)-dimensional Lorentzian manifold, whose metric