We define $F_{i}:S^{3}\times[i^{-1},1]$, sending $(x,t)$ to $\exp_{f_{i}(x)}((t-i^{-1})N(x))$, where $N(x)$ is the outward normal vector at $x$. Consider a Jacobi field $J(t)$ along a geodesic $\gamma_{x}(t)=F_{i}(x,t)$. Then since the curvature of $g$ is uniformly bounded, by Rauch comparison theorem there are constants $C_{1}>0$ and $\delta>0$ independent of $x$ and $i$ such that $C_{1}^{-1}|J(i^{-1})|_{g}\leq|J(t)|_{g}\leq C_{1}i|J(i^{-1})|_{g}$ for $t\in[i^{-1},\delta]$. For simplicity of notation we may assume $\delta=1$. So for $i$ large enough $F_{i}$ has no critical points in $[i^{-1},1]$. Indeed $F_{i}$ is a diffeomorphism. For otherwise there would be a geodesic loop $\sigma(s)(s\in[0,T])$ which is perpendicular to $S_{i}$ when $s=0$ and $s=T$. It is then easy to see this could not happen for sufficiently large $i$, by passing to a tangent cone.

Now we write $F_{i}^{*}g=dt^{2}+t^{2}h_{i}(t)$. First we notice that $|d_{g}(0,F_{i}(x,t))-t|\leq i^{-1}\epsilon_{i}$. Now we derive estimates for $g_{i}(t)$. Given a unit tangent vector $\xi$ at $x\in S^{3}$. Let $J(t)$ be the Jacobi field along $\gamma_{x}(t)$ with $J(i^{-1})=df_{i}(\xi)$ and $\dot{J}(i^{-1})=A_{S_{i}}(J(i^{-1}))$. Then $J(t)=d{F_{i}}_{(x,t)}(\xi)$. Clearly $||J(i^{-1})|_{g}-i^{-1}|\leq i^{-1}\epsilon_{i}$ and $|\dot{J}(i^{-1})-iJ(i^{-1})|_{g}\leq 2\epsilon_{i}$. Let $\{e_{1}(t),\cdots,e_{n}(t)=\dot{\gamma}_{x}(t)\}$ be an orthonormal frame of parallel vector fields along $\gamma_{x}(t)$, such that $J(i^{-1})=|J(i^{-1})|_{g}e_{1}$. Under the decomposition $J(t)=\sum_{\alpha}J_{\alpha}(t)e_{\alpha}(t)$ we have

$$\ddot{J}_{\alpha}(t)+\sum_{\beta}R_{\alpha n\beta n}(\gamma_{x}(t))J_{\beta}(t)=0,$$

where $R_{\alpha n\beta n}=R(e_{\alpha},e_{n},e_{\beta},e_{n})$. From the above discussion we have $|J(t)|\leq 2C_{1}$ for $t\in[i^{-1},1]$. So it is easy to see that there is a constant $C_{2}>0$ such that

$$||J(t)|_{g}-t|\leq C_{2}(i^{-1}+\epsilon_{i}t+t^{3}).$$

Thus

$$|h_{i}(t)-h_{0}|_{L^{\infty}_{h_{0}}}\leq C_{2}(i^{-1}t^{-1}+\epsilon_{i}+t^{2}).$$

Now take a unit tangent vector $X$ at $x$, we vary $J(i^{-1})$ so that $\nabla^{0}_{X}J(i^{-1})=0$ at $x$, and extend $X$ to a unit tangent vector field in a neighborhood $U$ of $x$ in $S^{3}$. We may also view $X$ as a tangent vector field on $U\times[i^{-1},1]$. Now we differentiate the Jacobi field equation, and similar arguments as above yield

$$|\nabla_{X}J(t)|_{g}\leq C_{3}(i^{-1}+\epsilon_{i}t+t^{3}),$$

for a constant $C_{3}>0$. This implies that there is a constant $C_{4}>0$ such that $t^{-1}|\nabla^{0}(h_{i}(t)-h_{0})|_{h_{0}}\leq C_{4}(i^{-1}t^{-2}+\epsilon_{i}t^{-1}+t).$ Similarly one can get bounds on higher derivatives of $h_{i}(t)-h_{0}$. The point is that for a fixed $\tau>0$ as $i$ goes to infinity we know $F_{i}(x,t)$ converges in $C^{3}$ to a limit $F^{\tau}_{\infty}(x,t)$ on $S^{3}\times[\tau,1]$. Then we can let $\tau\rightarrow 0$ and obtain a limit $F:S^{3}\times(0,1]$ with the property that $d_{g}(0,F(x,t))=t$, and

$$t^{-2}|F^{*}g-g_{0}|_{C^{0}_{g_{0}}}+t^{-1}|F^{*}g-g_{0}|_{C^{1}_{g_{0}}}+|F^{*}g-g_{0}|_{C^{2}_{g_{0}}}\leq C_{5},$$

for some constant $C_{5}>0$. This implies that $F^{*}g$ extends to a $C^{1,\alpha}$ metric on $B$. ∎