Manuscript X-G-ST@1 · X-G-ST.tex · section · 3c97567f

Fixed width and supports

Frozen operators, switches, and the predecessor interface › Fixed width and supports · lines 302–344

X-G-ST@1#eq:separation

Fix

m \geq 0, f \in C_{c}^{\infty} (O; R), g \in C_{c}^{\infty} ((0, T_{*}) \times R^{3}; [0, 1]) .

Let $U$ be a nonempty relatively compact open plateau:

U ⋐ L \cap int {g = 1}, g_{p} = C_{c}^{\infty} (U; T M) .

Write

K_{f} = supp f, S_{f} = supp g \cap J_{η}^{-} (K_{f}), C_{f} = K_{f} \cup S_{f}, D_{f} = J_{η}^{-} (K_{f}) \cap M_{T_{*}} .

The symbol $K$ , without a subscript, is reserved for the Killing operator.

Choose $δ > 0$ with

K_{f} \cup supp g \cup U \subset {2 δ < t < T_{*}}, 0 < ε < δ .

The onset $σ_{ε} (t) = χ (t / ε)$ is smooth, zero for $t \leq 0$ , one for $t \geq ε$ , and flat at both endpoints. The width is fixed. No limit $ε \to 0$ is taken.

The classical coefficients $h, k$ remain arbitrary smooth real symmetric tensors. In particular, they need not vanish in $0 < t < ε$ . The metric jets below are not updates of the host metric $G$ .

Displayed formulas in this passage

X-G-ST@1#eq:separation · line 327
$K_{f} \cup supp g \cup U \subset {2 δ < t < T_{*}}, 0 < ε < δ .$
unlabeled · line 305
$m \geq 0, f \in C_{c}^{\infty} (O; R), g \in C_{c}^{\infty} ((0, T_{*}) \times R^{3}; [0, 1]) .$
unlabeled · line 311
$U ⋐ L \cap int {g = 1}, g_{p} = C_{c}^{\infty} (U; T M) .$
unlabeled · line 317
$K_{f} = supp f, S_{f} = supp g \cap J_{η}^{-} (K_{f}), C_{f} = K_{f} \cup S_{f}, D_{f} = J_{η}^{-} (K_{f}) \cap M_{T_{*}} .$