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 $B(0,1)$.



A basic question about compact $C^1$ surfaces with boundary
By the classification of surfaces, $S$ is $C^1$ diffeomorphic to a closed disk (Morris Hirsch, Differential Topology, p. 205 theorem 3.7, for the $C^{\infty}$ case, combined with Whitney's smoothing theorem [p. 51 theorem 2.9] to get from $C^{\infty}$ to $C^1$). But any $C^1$ diffeomorphism $F \colon B \to S$ from the closed disk has locally bounded differential, in any Riemannian metric (because it is $C^1$), and so by compactness that bound has a maximum.


