Manuscript XI-A-ST@1 · XI-A-ST.tex · section · c3f0929d

Analytical freeze and the already-established baseline

Analytical freeze and the already-established baseline · lines 419–466

Work on

M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ = {t = 0},

as subsets of $M$ . Every coefficient field is real and smooth on a neighborhood of the closed slab.

Fix

m \geq 0, g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset .

Retain $0 < ε < T$ and

σ_{ε} (t) = χ (t / ε),

where

χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.

Choose the retained plateau domain

U ⋐ P_{g} \cap {ε < t < T} \cap V .

The admitted parameters are compactly supported in $U$ .

The plateau group $G_{P}$ consists of the corresponding orientation-preserving compact flows. For order bookkeeping, use genuine flows generated by $κ ξ + κ^{2} ζ$ , with $ξ, ζ \in C_{c}^{\infty} (U; T M_{T})$ . They preserve $g$ and $σ_{ε}$ and are the identity near $Σ$ .

Displayed formulas in this passage

unlabeled · line 423
$M_{T} = [0, T] \times R^{3}, M_{T}^{\circ} = (0, T) \times R^{3}, Σ = {t = 0},$
unlabeled · line 433
$m \geq 0, g \in C_{c}^{\infty} (M_{T}^{\circ}; [0, 1]), P_{g} = int {g = 1} \neq = \emptyset .$
unlabeled · line 440
$σ_{ε} (t) = χ (t / ε),$
unlabeled · line 444
$χ (s) = ⎩ ⎨ ⎧ 0, \frac{e ^{- 1/ s}}{e ^{- 1/ s} + e ^{- 1/ (1 - s)}}, 1, s \leq 0, 0 < s < 1, s \geq 1.$
unlabeled · line 454
$U ⋐ P_{g} \cap {ε < t < T} \cap V .$

Analytical freeze and the already-established baseline

Displayed formulas in this passage

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