Constrained harmonic data

Let ${\mathcal{S}}_{\mathrm{C}}$ consist of pairs with compact smooth Cauchy data obeying

These imply $G^{(1)}(h)=0$. We do not need a theorem placing every vacuum gauge orbit into this representative class.

Write

The velocity datum $B_{\mu \nu }$ is unrelated to $B_{\mathrm{P}}[h]$. Set $\bar{A}=\mathsf{r}A,\bar{B}=\mathsf{r}B$.

Let ${\mathcal{X}}_{\mathrm{C}}$ be the compact smooth tuples satisfying

. Choose radii $r_n\uparrow \infty$, each large enough to contain $\mathrm{supp}\chi$. Let ${{\mathcal{X}}_{\mathrm{C}}}_n$ consist of constrained tuples supported in ${\overline{\mathbb{B}}}_{r_n}$, with the inherited smooth Fréchet topology. Give

the locally convex inductive-limit topology.

Let ${{\mathcal{S}}_{\mathrm{C}}}_n$ consist of the solutions of (eq:C1harmonic) supported in

with the compact-open smooth topology. Give ${\mathcal{S}}_{\mathrm{C}}={\mathrm{indlim}}_n{{\mathcal{S}}_{\mathrm{C}}}_n$ its locally convex inductive-limit topology. All compact-data solutions belong to some stage.

Each stage is a closed Fréchet subspace of its ambient smooth space: support and differential equations are closed conditions. The inclusions of successive stages are closed and preserve the stage topology. These declarations avoid relying on an unspecified topology for the union of compact-data solutions.