Manuscript VI@1 · VI.tex · section · b5b60afb

A concrete compact test family

The finite obstruction certificate › A concrete compact test family · lines 2681–2719

The proof of Theorem thm:plateau-obstruction can be implemented with standard bump functions. Let

b (s) = {exp (- 1/ (1 - s^{2})), 0, ∣ s ∣ < 1, ∣ s ∣ \geq 1.

Let $B \in C_{c}^{\infty} (R^{4})$ be a nonnegative product of rescaled copies of $b$ , supported in the unit coordinate cube and normalized by

\int_{R^{4}} B (y) d^{4} y = 1.

For a point $x_{*} = (t_{*}, 0)$ , put

f_{ϵ} (x) = ϵ^{- 4} B ((x - x_{*}) / ϵ) .

Choose $ϵ > 0$ small enough that

supp f_{ϵ} \subset U, \partial_{1} ϕ_{0} \geq \frac{1}{2} on U .

Smooth plateau cutoffs give $g$ and $χ_{ξ}$ with

g = 1 near supp f_{ϵ}, χ_{ξ} = 1 near supp g .

All supports can be kept inside $(0, T) \times R^{3}$ .

This construction is exact. It does not claim rational values for the bump integrals.

Displayed formulas in this passage

unlabeled · line 2685
$b (s) = {exp (- 1/ (1 - s^{2})), 0, ∣ s ∣ < 1, ∣ s ∣ \geq 1.$
unlabeled · line 2695
$\int_{R^{4}} B (y) d^{4} y = 1.$
unlabeled · line 2699
$f_{ϵ} (x) = ϵ^{- 4} B ((x - x_{*}) / ϵ) .$
unlabeled · line 2703
$supp f_{ϵ} \subset U, \partial_{1} ϕ_{0} \geq \frac{1}{2} on U .$
unlabeled · line 2710
$g = 1 near supp f_{ϵ}, χ_{ξ} = 1 near supp g .$