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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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  • $\begingroup$ Is $S$ a hypersurface in Euclidean space? $\endgroup$
    – Ben McKay
    Oct 24, 2019 at 19:26
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    $\begingroup$ What is $S$ is an annulus, not $C^1$ diffeomorphic to a closed ball? $\endgroup$
    – Ben McKay
    Oct 24, 2019 at 19:28
  • $\begingroup$ When you ask about constructing such a map, what data do you provide? For example, do you know $S$ as a Riemannian manifold, with an explicit metric, and then I have to find $F$? $\endgroup$
    – Ben McKay
    Oct 24, 2019 at 19:59
  • $\begingroup$ Your question is rather vague, and appears to be using terminology different than what people use in differential topology/geometry. Generally co-dimension one submanifolds are not discs or balls, if that's what your question is concerned with. $\endgroup$
    – Ryan Budney
    Oct 24, 2019 at 20:06
  • $\begingroup$ @BenMcKay Yes, you are right. Assume that $S$ is diffeomorphic to a closed ball. $\endgroup$
    – MathLearner
    Oct 24, 2019 at 22:23

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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.

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