Dominant energy condition and spinors on Lorentzian manifolds

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 $\alpha$-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 $\pi_1(M)$ 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 $\phi$ is in the kernel of the Dirac--Witten operator on $(M,g,K)$ if and only if $(M,g,K,\phi)$ admits an extension to a Lorentzian manifold $(\overline N,\overline h)$ with parallel spinor $\bar\phi$ 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 $\phi$ is the restriction of $\bar\phi$ to $M$.