Bartnik's Hilbert-manifold conjecture for the constraint map

Let $\mathcal{F}$ be the space of initial data satisfying the prescribed boundary conditions, with $\mathcal{G}$ the space of metrics and $\mathcal{K}$ the space of momentum tensors. Let

$$\Phi:\mathcal{F}\rightarrow\mathcal{N}$$

be the constraint map into a suitable Hilbert space $\mathcal{N}$. Bartnik's Hilbert-manifold conjecture. For some choice of $\mathcal{N}$, $\Phi$ is a smooth map of Hilbert manifolds and

$$D\Phi(g,\pi)$$

is surjective at every point $(g,\pi)\in\mathcal{G}\times\mathcal{K}$. Consequently, the level sets of $\Phi$ are Hilbert submanifolds of $\mathcal{F}$. The conjecture is motivated by Bartnik's phase-space argument; the source states that the required surjectivity is the unresolved step in the presence of a boundary and intends to pursue a proof.