The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds

Question

(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.)

Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian manifold with a symmetric 2-tensor field $k\in \Gamma(T^2 T^*\Sigma)$. The Cauchy problem in general relativity asks for sufficient conditions for $\Sigma$ to be isometrically embedded in an $(n+1)$-dimensional Lorentzian manifold $(M,g)$ with $Rc(g) =0$ such that $k$ becomes the scalar second fundamental form on $\Sigma$ when viewed as a submanifold of $M$. Such a manifold $(M,g)$ is called an Einstein development of $(\Sigma,\bar{g},k)$. The Einstein constraint equations \begin{align*} S(\bar{g}) -\lvert k\rvert_{\bar{g}}^2 + (\mathrm{tr}_{\bar{g}}h)^2 &= 0,\\ \mathrm{div}_{\bar{g}}k - d(\mathrm{tr}_{\bar{g}}k)&=0 \end{align*} (where $S(\bar{g})$ is the scalar curvature of $\Sigma$) are necesary conditions for this to be the case. Choquet-Bruhat proved in the 1950s that the Einstein constraints along with a few additional regularity assumptions are also sufficient.

I have been reading Choquet-Bruhat's treatment of the subject in General Relativity and the Einstein Equations. As I don't know too much analysis or PDE theory, there is a theorem on quasilinear hyperbolic PDEs which I've had to take as a black box. In order to state this theorem, we put an additional "background" Riemannian metric $e$ on $\Sigma$. Define the metric $\hat{e}=e+dt^2$ on $\Sigma\times \mathbb{R}$. In addition, let $\hat{\nabla}$ denote covariant differentiation with respect to $\hat{e}$ on $\Sigma\times \mathbb{R}$. Choquet-Bruhat requires that $(\Sigma,e)$ is Sobolev regular, so that the Sobolev spaces on $\Sigma$ defined with respect to $e$ satisfy the following properties:

• $H_s(\Sigma,e)$ continuously embedds into the space of continuous and bounded functions on $\Sigma$ if $s>n/2$,
• Multiplication defines a continuous map $H_{s_1}(\Sigma,e)\times H_{s_2}(\Sigma,e)\to H_s(\Sigma,e)$ if $s_1+s_2>s+n/2$ and $s_1,s_2\geq s$.

The theorem then takes the following form (although specific details may be a little off): Under some regularity conditions, an equation of the form $$g^{\alpha\beta}(.,u,\hat{\nabla}u) \hat{\nabla}^2_{\alpha\beta}u + F(.,u,\hat{\nabla}u)=0 \qquad \text{on } \Sigma\times\mathbb{R}$$ (where $u$ is the unknown and $g^{\alpha\beta}$ is a Lorentzian metric which may depend on $u$ and $\hat{\nabla} u$) has a solution on some strip $\Sigma\times [0,T]$, taking prescribed initial values $$u=\varphi \in H_s(\Sigma,e)\quad\text{and}\quad \partial_t u = \psi \in H_{s-1}(\Sigma,e)\quad\text{on }\Sigma\times \{0\},\quad s>n/2+1.$$

I have tried looking at some other references on quasilinear second order hyperbolic PDEs, but outside of Appendix III in Choquet-Bruhat's book, all of them seem to only consider the case in which $\Sigma$ is diffeomorphic to an open subset of $\mathbb{R}^n$. I have the following questions:

1. Is the above theorem for quasilinear hyperbolic PDEs well-established for general $\Sigma$? Are there multiple references which cover this theorem for general $\Sigma$ outside of Appendix III in Choquet-Bruhat's book?

2. Are there references for Sobolev regular Riemannian metrics? From my searching online, it seems this terminology is not widely used. Choquet-Bruhat doesn't give a proof for the existence of such metrics, but mentions (without proof) that complete Riemannian manifolds and Riemannian manifolds with Lipschitzian boundary (I am not sure what this last term means) are Sobolev regular.

3. What are the precise requirements for the initial data $(\Sigma,\bar{g},k)$ to admit an Einstein development? One usually asks for $(\bar{g},k)\in H^s(\Sigma)\times H^{s-1}(\Sigma)$ for some value of $s$, but I am not so much concerned here with the optimal value of $s$. Rather, I am concerned with other regularity assumptions which seem to vary from paper to paper. Choquet-Bruhat's book and this paper by Choquet-Bruhat, Christodoulu & Francaviglia both require the existence of a Sobolev regular metric $e$ on $\Sigma$ with respect to which $\bar{g}$ is uniformly equivalent in some sense. The latter paper also requires $e$ to be asymptotically flat. Other sources make no such mention of a background metric, but then only consider the case where $\Sigma$ is diffeomorphic to $\mathbb{R}^n$, e.g. this paper by Hughes, Kato & Marsden. Some additionally impose an asymptotic flatness condition on $(\bar{g},k)$, e.g. this paper by Fischer and Marsden. Still others impose no additional constraints on the data $(\Sigma,\bar{g},k)$ other than the Einstein constraint equations, such as Ringström's book The Cauchy Problem in General Relativity.

Answer 1

Have a look at Ringström's The Cauchy Problem in General Relativity (EMS 2009). He spends several chapters building up the analytical material of the kind that you are asking about.