There is a sequence $\epsilon_{i}\rightarrow 0$ and a sequence of smooth embeddings $f_{i}$ from $S^{3}$ to $B$ with the properties

1. (1)

   $d_{GH}(S_{i},\partial B(i^{-1}))\leq i^{-1}\epsilon_{i}$ where $S_{i}=f_{i}(S^{3})$.

2. (2)

   $|i^{2}f_{i}^{*}g-h_{0}|_{C^{4}_{h_{0}}}\leq\epsilon_{i}$, where $h_{0}$ is the standard round metric on $S^{3}$.

3. (3)

   $|i^{-1}A_{S_{i}}+Id|_{C^{3}_{h_{0}}}\leq\epsilon_{i},$ where $A_{S_{i}}:TS_{i}\rightarrow TS_{i}$ is the shape operator.