Switch, regulator, plateau, gauge group

Fix

Fix the ramp

which is smooth, nondecreasing, and satisfies ${\chi }^{\prime }(s)>0$ for $0<s<1$: with $a=e^{-1/s}$, $b=e^{-1/(1-s)}$, ${\chi }^{\prime }=(a^{\prime }b-a{b}^{\prime })/(a+b{)}^2$, $a^{\prime }=a/s^2>0$, $b^{\prime }=-b/(1-s{)}^2<0$. Fix $0<\epsilon <T$ and

Then ${\sigma }_{\epsilon }=0$ on $\{t\le 0\}$, ${\sigma }_{\epsilon }=1$ on $\{t\ge \epsilon \}$, ${\partial }_{\mu }{\sigma }_{\epsilon }={{\delta }^0}_{\mu }{\sigma }_{\epsilon }^{\prime }$ with ${\sigma }_{\epsilon }^{\prime }(t)={\epsilon }^{-1}{\chi }^{\prime }(t/\epsilon )$, and ${\sigma }_{\epsilon }^{\prime }>0$ exactly on $\{0<t<\epsilon \}$.

Fix a nonempty open, relatively compact $U$ with

and set

It is a Lie algebra under the bracket of vector fields. For $\xi ,\zeta \in {\mathfrak{g}}_U$ let $V_{\kappa }=\kappa \xi +{\kappa }^2\zeta$ and let $F_{\kappa }={\mathrm{Fl}}_{V_{\kappa }}^1$ be its time-one flow, a diffeomorphism of $M$ for every real $\kappa$.