Manuscript XIV-E@1 · XIV-E.tex · input · e89fd2a5

Input statement: XIV-C: characteristic geometry

Release criteria and copied premises · lines 375–412

XIV-E@1#input:XIVC

The retained triple is

u_{0} = (t_{*}, 0, 0, 0), y_{0} = (t_{*} + L, L, 0, 0), x_{0} = (t_{*} + 2 L, 2 L, 0, 0),

with $ε < t_{*} < t_{*} + 2 L < T$ , $f (x_{0}) \neq = 0$ , and $g = 1$ near these points.

In the order $(b, a, y) = (x, u, y)$ , the characteristic stratum is

N^{\circ} = {y = a + α ℓ_{n}, b = a + β ℓ_{n}, ℓ_{n} = (1, n), n \in S^{2}, 0 < α < β} .

It is smooth of codimension four. For

q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2},

the differentials have rank two on $N^{\circ}$ and satisfy

- \frac{d q _{R}}{2 β} + \frac{d q _{1}}{2 α} + \frac{d q _{2}}{2 ( β - α )} = 0.

No tangent vector makes the three selected oriented evaluations strictly positive.

Collision-supported changes vanish on the retained collision-free neighborhood, while the original attachment variables are retained XIVC,XIVD.

Displayed formulas in this passage

unlabeled · line 378
$u_{0} = (t_{*}, 0, 0, 0), y_{0} = (t_{*} + L, L, 0, 0), x_{0} = (t_{*} + 2 L, 2 L, 0, 0),$
unlabeled · line 387
$N^{\circ} = {y = a + α ℓ_{n}, b = a + β ℓ_{n}, ℓ_{n} = (1, n), n \in S^{2}, 0 < α < β} .$
unlabeled · line 397
$q_{R} = (b - a)^{2}, q_{1} = (y - a)^{2}, q_{2} = (b - y)^{2},$
unlabeled · line 401
$- \frac{d q _{R}}{2 β} + \frac{d q _{1}}{2 α} + \frac{d q _{2}}{2 ( β - α )} = 0.$