Manuscript X-J-ST@1 · X-J-ST.tex · section · 8f0d1a36

Finite polarization of the retained vertex

Section A: two action-derived compact charged coefficients › Finite polarization of the retained vertex · lines 674–695

X-J-ST@1#eq:weights

The open set ${a > 0}$ is nonempty. Choose two disjoint relatively compact balls in this set and nonzero nonnegative functions

u_{i} \in C_{c}^{\infty} (U), i = 1, 2,

supported in those balls. Define

M_{i} = \int a u_{i}^{3} d^{4} x > 0, w_{i} = \frac{a u _{i} ^{3}}{M _{i}}, \int w_{i} d^{4} x = 1.

Their supports $K_{i} = supp w_{i}$ are disjoint. All choices occur inside the already fixed plateau.

Freeze the calibration scalar $ϕ_{cal} (x) = x^{1}$ . It is an off-shell smooth function, not an assumed solution.

Displayed formulas in this passage

X-J-ST@1#eq:weights · line 683
$M_{i} = \int a u_{i}^{3} d^{4} x > 0, w_{i} = \frac{a u _{i} ^{3}}{M _{i}}, \int w_{i} d^{4} x = 1.$
unlabeled · line 679
$u_{i} \in C_{c}^{\infty} (U), i = 1, 2,$