# A basic question about compact $C^1$ surfaces with boundary

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 $$B(0,1)$$.

• Is $S$ a hypersurface in Euclidean space? Oct 24, 2019 at 19:26
• What is $S$ is an annulus, not $C^1$ diffeomorphic to a closed ball? Oct 24, 2019 at 19:28
• 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 $F$? Oct 24, 2019 at 19:59
• 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. Oct 24, 2019 at 20:06
• @BenMcKay Yes, you are right. Assume that $S$ is diffeomorphic to a closed ball. Oct 24, 2019 at 22:23

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.