Moser stability theorem

Put ${\alpha }_t={\omega }_t-{\omega }_0$. Constancy of the de Rham class says each ${\alpha }_t$ is exact. Although $[0,1]$ is not a boundaryless parameter manifold, the proof of [F1] constructs one fixed linear primitive operator from a finite good cover, finite spatial homotopy integrals, a finite-dimensional linear solver, and a fixed partition of unity. Apply that same operator pointwise to ${\alpha }_t$. Every one of its finite operations preserves all one-sided time derivatives and joint spatial smoothness at the closed endpoints, so ${\lambda }_t=R{\alpha }_t$ is smooth up to $t=0,1$ and satisfies $d{\lambda }_t={\alpha }_t$. Put ${\sigma }_t={\dot{\lambda }}_t$; differentiating gives $d{\sigma }_t={\dot{\omega }}_t$. By [F2], the equations ${\iota }_{X_t}{\omega }_t=-{\sigma }_t$ have a unique jointly smooth solution $X_t$ up to both endpoints.

We first put the field on a genuinely open time interval without assuming an extension of the forms. Take the step $\tau$ from [F4]. On $0<s<1$ its defining quotient has positive derivative, since $\frac{d}{ds}\log (\beta (s)\mathrm{/}\beta (1-s))=s^{-2}+(1-s{)}^{-2}>0$; hence it maps $(0,1)$ diffeomorphically onto $(0,1)$. Define $Y_s={\tau }^{\mathrm{\prime }}(s)X_{\tau (s)}$ for $0<s<1$ and $Y_s=0$ outside. Every derivative of $\tau$ is flat at $0,1$ by [F4], while all one-sided mixed derivatives of $X_t$ from step 1.1 are continuous on compact $[0,1]\times M$. The product rule therefore shows that $Y_s$ is a smooth time-dependent field on the open interval $\mathbb{R}$. Apply [F3] to $Y$. Fix the Riemannian metric used in [F1]'s construction; $\parallel Y_s\parallel$ is bounded on $[0,1]\times M$. The distance along a trajectory between times $r,s$ is at most its length and at most $C\mathrm{\mid }r-s\mathrm{\mid }$, so a finite-time maximal trajectory is Cauchy. Compactness gives its limit, and [F3] at that interior time extends it. Thus the evolution ${\psi }_s$ exists through $s=1$, with inverse given by reverse evolution. For $0<t<1$ set ${\varphi }_t={\psi }_{{\tau }^{-1}(t)}$, with ${\varphi }_0=\mathrm{id}$ and ${\varphi }_1={\psi }_1$. Changing variables in the coordinate integral equation for $\psi$ shows that, on each short time interval whose trajectory lies in one chart, ${\varphi }_t(p)={\varphi }_{t_0}(p)+{\int }_{t_0}^t{X}_u({\varphi }_u(p))du$ in that chart. The integral equation and the up-to-endpoint smoothness of $X_t$ bootstrap ${\varphi }_t$ and its spatial derivatives to joint smoothness in $(t,p)$ through both endpoints, despite the nonsmooth inverse of $\tau$ there. Each ${\varphi }_t$ is a diffeomorphism, with inverse from the reverse evolution.

The curve ${\varphi }_t$ from step 2.1 is the evolution of $X_t$ in the original time parameter. The pullback equation in [F2] and step 1.1 give $\frac{d}{dt}({\varphi }_t^{\ast }{\omega }_t)=0$, hence ${\varphi }_t^{\ast }{\omega }_t={\varphi }_0^{\ast }{\omega }_0={\omega }_0$ for the entire closed interval. Empty $M$ uses the empty isotopy.