Let $S \subset \mathbb{R}^3$ be a compact and locally $C^1$ simply-connected surface with a $C^1$ boundary with no self intersection. Is there a $C^1$ bijection $F: \overline{B(0,1)} \rightarrow \overline{S}$ such that $ a|\xi|^2 \leq (D F)_{i,j} \xi_i \xi_j \leq A |\xi|^2$ (a,A>0), $F$ maps $\partial B(0,1)$ to $\partial S$, and $F^{-1}$ is also a $C^1$ map? How such a map could be reconstructed? I would appreciate a reference.

Basically, I am looking for a nice differentiable map between $S$ and $B(0,1)$.

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