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arXiv 2103.11032math.DG
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洛伦兹流形上的优势能量条件和旋量

Dominant energy condition and spinors on Lorentzian manifolds

Bernd Ammann, Jonathan Glöckle

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摘要：

设$(\overline M,\overline g)$是时空定向的洛伦兹自旋流形，$M$是$\overline M$的紧致类空超曲面，具有诱导黎曼度量$g$和第二基本形式$K$ .如果$(\overline M,\overline g)$满足严格意义上的优势能量条件，则$M\subseteq \overline M$的Dirac-Witten算子是可逆的自伴Fredholm算子.这使得我们可以使用指标理论的方法，以检测非平凡同伦群的初始空间中的$M$满足在严格意义上的主导能量条件。中心工具将是希钦的$α$不变的洛伦兹模拟。当主能量条件仅在弱意义下成立时，Dirac-Witten算子可能是不可逆的，我们将研究这种算子的核。我们将表明，核可能是非平凡的，如果$π_1(M)$是几乎可解的派生长度在最多$2$。这允许扩展的索引理论方法的初始数据的空间，满足在弱意义上的主导能量条件。我们将进一步证明旋量$ϕ$在$(M,g,K)$上Dirac-Witten算子的核中当且仅当$(M,g,K,ϕ)$允许Lorentz流形$(\overline N,\overline h)$的扩张具有平行旋量$\barϕ$，使得$M$是$(\overline N,\overline h)$的Cauchy超曲面，使得$g$和$K$分别是$M$的导出度量和第二基本形式，并且$ϕ$是$\barϕ$对$M$的限制。

英文摘要：

Let $(\overline M,\overline g)$ be a time- and space-oriented Lorentzian spin manifold, and let $M$ be a compact spacelike hypersurface of $\overline M$ with induced Riemannian metric $g$ and second fundamental form $K$. If $(\overline M,\overline g)$ satisfies the dominant energy condition in a strict sense, then the Dirac--Witten operator of $M\subseteq \overline M$ is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial on $M$ satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's $α$-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac--Witten operator may be non-invertible, and we will study the kernel of this operator in this case. We will show that the kernel may only be non-trivial if $π_1(M)$ is virtually solvable of derived length at most $2$. This allows to extend the index theoretical methods to spaces of initial data, satisfying the dominant energy condition in the weak sense. We will show further that a spinor $ϕ$ is in the kernel of the Dirac--Witten operator on $(M,g,K)$ if and only if $(M,g,K,ϕ)$ admits an extension to a Lorentzian manifold $(\overline N,\overline h)$ with parallel spinor $\barϕ$ such that $M$ is a Cauchy hypersurface of $(\overline N,\overline h)$, such that $g$ and $K$ are the induced metric and second fundamental form of $M$, respectively, and $ϕ$ is the restriction of $\barϕ$ to $M$.

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