Relative transversality for an embedded surface with fixed boundary

At every $x\in \mathrm{\partial }F$, the inclusion $T_x\mathrm{\partial }F\subset T_{x}F$ and the boundary transversality hypothesis imply $d{i}_x(T_{x}F)+T_{i(x)}S=T_{i(x)}M$. Thus $i$ is transverse to $S$ along $\mathrm{\partial }F$. Transversality is open in the point of the source, and $\mathrm{\partial }F$ is compact, so there is a collar neighbourhood $U$ of $\mathrm{\partial }F$ on which $i$ is transverse to $S$.

Choose a smooth collar coordinate $s$ on $U$ and a smooth function $\gamma :F\to [0,1]$ that vanishes on a smaller boundary collar and equals $1$ outside $U$. Such a function is obtained by composing the collar coordinate with a smooth one-variable cutoff and extending by $1$. Put $\tau ={\gamma }^2$. Then $d\tau =0$ wherever $\tau =0$, and the zero set of $\tau$ lies inside $U$.

Apply [F1] to the boundaryless interior $F^{\circ }$ and the map $i{\mathrm{\mid }}_{F^{\circ }}$, obtaining a smooth family $\mathcal{F}:F^{\circ }\times B\to M$ with $\mathcal{F}(p,0)=i(p)$ and each parameter map a submersion. On $F^{\circ }$ define $G(p,a)=\mathcal{F}(p,\tau (p)a)$, restricting $B$ if necessary so $\tau (p)a\in B$. Where $\tau >0$, the parameter derivative of $G$ is surjective. Where $\tau =0$, $d\tau =0$, so $G(p,a)=i(p)$ and its source derivative is $d{i}_p$; [F5] and step 1.1 make it transverse to $S$. Since $\tau =0$ on a boundary collar, setting $G=i$ on $\mathrm{\partial }F$ extends $G$ smoothly to $F\times B$, fixed near the boundary. Thus the total family is transverse to $S$.

Apply [F2] to $G{\mathrm{\mid }}_{F^{\circ }\times B}$. Its good parameters are dense, because the bad set is null and every open parameter ball has positive measure. For each good parameter $a$, $G_a$ is transverse on $F^{\circ }$; near $\mathrm{\partial }F$ it equals $i$, which is transverse by step 1.1. Hence $G_a$ is transverse on all of $F$.

Choose compatible Riemannian metrics on $F$ and $M$ using [F6]. At each $x\in F$, choose coordinate charts and a linear projection $P_x$ to a two-dimensional tangent plane so that $D(P_x\circ i)$ at $x$ is invertible. By [F5], shrink to a convex coordinate neighbourhood $U_x$ on which this derivative stays within one quarter of the least singular value of its value at $x$. Every sufficiently $C^1$-close map $j$ then has $P_x\circ j$ injective on $U_x$: integrate its derivative along the coordinate segment between two points to obtain a lower bound of one half that least singular value times their coordinate distance. Compactness gives a finite cover by such $U_x$ and a Lebesgue number $\delta >0$. For pairs at source distance at least $\delta$, compactness and injectivity of $i$ give a positive minimum target distance, which persists under sufficiently small $C^0$ perturbations. The differential of $i$ has a positive minimum norm on the compact unit tangent bundle, so immersion also persists under sufficiently small $C^1$ perturbations. Thus all sufficiently $C^1$-close maps are injective immersions; compactness of $F$ and the Hausdorff property of $M$ make them embeddings. Smoothness of $G$ and compactness of $F$ imply $G_b\to i$ in $C^1$ as $b\to 0$, so there is a parameter ball about $0$ for which every $G_b$ is an embedding. By the density in step 4.1 choose a good $a$ in that ball; then $t\mapsto G_{ta}$ is an isotopy through embeddings fixed near $\mathrm{\partial }F$. Apply [F4] to extend it to an ambient isotopy. Its endpoint carries $i(F)$ to a surface transverse to $S$, as required.