Manuscript X-H-ST@1 · X-H-ST.tex · section · 943aa27e

The retained host and the free analytical interface

The retained host and the free analytical interface · lines 319–351

The defect is a worldsheet, not a worldline. No host-update map, second cone, or global conical Green inverse is introduced IB,XAST,XGST.

Fix the compact scalar switch $g$ , an observation test $f$ , a plateau $U$ , and the positive onset width:

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

σ_{ε} (t) = χ (t / ε), 0 < ε < δ, supp f \cup supp g \cup U \subset {2 δ < t < T_{*}} .

Here $χ$ is smooth, zero on $(- \infty, 0]$ , one on $[1, \infty)$ , and flat at the endpoints. Thus $σ_{ε} = g = 1$ near $U$ . There is no shrinking-layer argument.

For compact supports 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_{*}} .

Displayed formulas in this passage

unlabeled · line 327
$U ⋐ L \cap int {g = 1}, g_{p} = C_{c}^{\infty} (U; T M),$
unlabeled · line 333
$σ_{ε} (t) = χ (t / ε), 0 < ε < δ, supp f \cup supp g \cup U \subset {2 δ < t < T_{*}} .$
unlabeled · line 345
$K_{f} = supp f, S_{f} = supp g \cap J_{η}^{-} (K_{f}), C_{f} = K_{f} \cup S_{f}, D_{f} = J_{η}^{-} (K_{f}) \cap M_{T_{*}} .$

Nearby in this section

assumption: Assumption: Unchanged supported presentation