Manuscript Em01-G@5 · Em01-G-v5.tex · section · 3fbfe951

Support isolation and a fixed complete-family detecting test

Support isolation and a fixed complete-family detecting test · lines 1754–1786

Em01-G@5#eq:witness-test

Fix such an isolated positive neighborhood $O$ . Let

q = r^{2} = (z - a)^{2} .

Near the non-tip positive cone use the smooth residual coordinates

ϵ = - r^{0}, ω = - \frac{r}{∣ r ∣}, q = r^{2} .

They form local coordinates: the spatial radius is recovered from $∣ r ∣^{2} = ϵ^{2} - q$ .

Let

η_{J} = (e^{J})^{μν} η_{μν}, J \sum η_{J} e_{J} = η .

Choose a compact smooth $ψ (ϑ, ϵ, ω)$ in a small tangential subchart, and a compact normal cutoff $ζ (u, v, ξ, q)$ , equal one near zero. Choose their supports so that the following test is supported in $O$ :

(Φ_{*})_{J} = η_{J} \frac{q ^{2}}{2} \frac{∣ ξ ∣ ^{2}}{2} ψ (ϑ, ϵ, ω) ζ (u, v, ξ, q) .

Displayed formulas in this passage

Em01-G@5#eq:witness-test · line 1779
$(Φ_{*})_{J} = η_{J} \frac{q ^{2}}{2} \frac{∣ ξ ∣ ^{2}}{2} ψ (ϑ, ϵ, ω) ζ (u, v, ξ, q) .$
unlabeled · line 1758
$q = r^{2} = (z - a)^{2} .$
unlabeled · line 1762
$ϵ = - r^{0}, ω = - \frac{r}{∣ r ∣}, q = r^{2} .$
unlabeled · line 1770
$η_{J} = (e^{J})^{μν} η_{μν}, J \sum η_{J} e_{J} = η .$

Support isolation and a fixed complete-family detecting test

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