Smooth relative isotopy extension for finite disk arc systems

Extend the track and cut off its velocity. Since $F$ is smooth on the compact square, extend it as an ${\mathbb{R}}^2$-valued smooth map $\bar{F}$ to an open rectangle containing $[0,1{]}^2$. Shrink the rectangle so that each slice $u\mapsto \bar{F}(u,s)$ remains an embedding on a slightly larger closed interval for $s$ in a neighborhood of $[0,1]$; this follows from ${\mathrm{\partial }}_{u}F\mathrm{\ne }0$ on the compact square and uniform separation of pairs of arc parameters away from the diagonal. Then $\widehat{F}(u,s):=(\bar{F}(u,s),s)$ is an injective immersion on that open rectangle. On a smaller compact rectangle it is a continuous injection into the Hausdorff space ${\mathbb{R}}^2\times \mathbb{R}$, hence an embedding; its restriction to the interior is an embedded surface $\mathrm{\Sigma }$ without boundary. Define the smooth field along it by $W(\widehat{F}(u,s)):=({\mathrm{\partial }}_s\bar{F}(u,s),0)$. The compact set $K_0:=\widehat{F}([\delta ,1-\delta ]\times [0,1])$ is disjoint from the closed set $B:=(\mathrm{\partial }{D}^2\cup P\cup C)\times \mathbb{R}$, by hypotheses 1 and 3. The extension lemma [L1] gives an open neighborhood $N$ of $\mathrm{\Sigma }$ and a smooth field $\tilde{W}$ on $N$ restricting to $W$. Choose an open $U_0$ with compact closure contained in $N\setminus B$ and containing $K_0$. By [L2] choose a smooth $\rho :{\mathbb{R}}^2\times \mathbb{R}\to [0,1]$ equal to $1$ near $K_0$ and supported in $U_0$. The field $V:=\rho \tilde{W}$ on $N$, extended by zero outside $N$, is smooth and compactly supported. Its spatial component $X_s(x):={\mathrm{pr}}_{{\mathbb{R}}^2}V(x,s)$ is a smooth time-dependent field on ${\mathbb{R}}^2$ whose support over $s\in [0,1]$ lies in a common compact subset of $\mathrm{int}{D}^2\setminus (P\cup C)$.

The stationary collars are fixed. Hypothesis 2 gives ${\mathrm{\partial }}_{s}F(u,s)=0$ for $u\in [0,\delta ]\cup [1-\delta ,1]$ and $s\in [0,1]$. At each such track point $\tilde{W}=W=0$, so $X_s(F(u,s))=0$. The constant curve at $F(u,0)$ therefore solves the flow equation; uniqueness gives ${\mathrm{\Psi }}_{s,0}(F(u,0))=F(u,0)=F(u,s)$ on both endpoint collars.

The flow fixes the required sets and preserves the disc. The support of $X$ lies in a compact subset of $\mathrm{int}{D}^2\setminus (P\cup C)$, so $X$ vanishes on a neighborhood of $\mathrm{\partial }{D}^2\cup P\cup C$. Uniqueness makes each of these points stationary under the flow, and no flow line crosses the boundary; thus every flow map carries $D^2$ onto itself and fixes $\mathrm{\partial }{D}^2$, $P$, and $C$ pointwise. Each ${\mathrm{\Psi }}_{s,0}$ is smooth with inverse ${\mathrm{\Psi }}_{0,s}$, hence a diffeomorphism of $D^2$; it is the identity for $s=0$.

The flow realizes the moving part. Let ${\mathrm{\Psi }}_{s,0}$ be the global evolution operator of $X$ over $[0,1]$, which exists since its supports lie in a common compact set. Fix $u\in [\delta ,1-\delta ]$ and put $\gamma (s):=F(u,s)$. At $\widehat{F}(u,s)\in K_0$ one has $\rho =1$, so the spatial component satisfies $X_s(\gamma (s))={\mathrm{\partial }}_{s}F(u,s)={\gamma }^{\mathrm{\prime }}(s)$ for every $s\in [0,1]$. Thus $\gamma$ solves the flow equation with $\gamma (0)=F(u,0)$, and uniqueness gives ${\mathrm{\Psi }}_{s,0}(F(u,0))=F(u,s)$. For $u$ outside this interval step 2.1 gives the same equality. Hence ${\mathrm{\Psi }}_{s,0}(F(u,0))=F(u,s)$ for all $u\in [0,1]$.

Conclusion and finite composition. Setting ${\mathrm{\Phi }}_s:={\mathrm{\Psi }}_{s,0}{\mathrm{\mid }}_{D^2}$ gives the smooth isotopy $\mathrm{\Phi }:D^2\times [0,1]\to D^2$ of the statement with ${\mathrm{\Phi }}_0=\mathrm{id}$, the pointwise stabilisations of step 2.2, and ${\mathrm{\Phi }}_s(F(u,0))=F(u,s)$ for every admissible pair by step 3.1. Moreover step 3.1 makes the whole construction available for each member of a finite sequence of such data, and the map $(s,x)\mapsto {\mathrm{\Psi }}_{s,0}(x)$ is smooth by [L3] and [L4]; a finite composite of these smooth isotopies again begins at the identity, fixes $\mathrm{\partial }{D}^2$, $P$ and $C$ pointwise at every time, and realizes the finite sequence of moves, which proves the final clause as well.