Misinterpretations of Hilbert's Theorem?

Question

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance:

"I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ for any $k$."

"However, as Hilbert showed us, the reverse is not true; we cannot embed the hyperbolic plane into Euclidean 3-space."

"$H^2$ does not isometrically embed in $R^3$ (Hilbert's theorem)."

However, these should only apply to smooth ($C^\infty$) embeddings. Nash-Kuiper, particularly, guarantees the existence of an isometric $C^1$ embedding, since the Klein disk model is a short map from $H^2 \to \Bbb R^3$.

This was noted in the answer to this MO question.

There's also this answer to the same question, which says there's no $C^2$ embedding.

But then on the other hand, this page on hyperbolic crochet at Cornell claims that the crochet models are not $C^1$ and can be extended indefinitely, which would appear to violate Hilbert's theorem.

At this point, I'm very confused about what is and isn't allowed regarding isometric embeddings of $H^2$ into $\Bbb R^3$. Obviously no infinitely differentiable isometric embedding exists, but there are obviously $C^1$ embeddings of the whole space. Furthermore, there are also isometric embeddings of compact subsets of $H^2$ into $R^3$, such as the crochet models, and it's not clear what differentiability restrictions there are for that.

Is there some way to better understand the slew of somewhat-contradictory statements made above? What, exactly, is and isn't allowed regarding isometric embeddings of $H^2$ into $R^3$?

Answer 1

1. A close look at the hyperbolic crochet article that you linked reveals very little mathematical content to it. The link does not even propose a Lipschitz isometric embedding of the hyperbolic plane into $E^3$. This might be a good mathematical PR article but not more than that. No wonder, one gets confused trying to read it. Can this paper be made precise and yield an actual mathematical theorem? Who knows (the paper is by now 16 year old), until then I suggest it should not be used as a reference.

2. The first proof of nonexistence of a $C^2$-smooth isometric immersion of a hyperbolic plane into $E^3$ is due to Efimov:

   N. V. Efimov, Impossibility of a complete regular surface in Euclidean 3-Space whose Gaussian curvature has a negative upper bound, Doklady Akad. Nauk. SSSR 150 (1963), 1206–1209 (Russian); Engl. transl. in Sov. Math. (Doklady) 4 (1963), 843–846.

   N. V. Efimov, Differential criteria for homeomorphism of certain mappings with application to the theory of surfaces, Mat. Sb., Nov. Ser. 76 (1968) (Russian); Engl. transl. in Math. USSR (Sbornik) 5 (1968), 475–488.

In fact, he proves an even stronger result:

Theorem. Suppose that $M$ is a simply connected complete Riemannian surface of sectional curvature $\le k<0$. Then there is no $C^2$-smooth isometric immersion of $M$ into $E^3$.

This is a vast generalization of Hilbert's theorem on nonexistence of $C^\infty$-smooth isometric immersions of the hyperbolic plane into $E^3$.

3. The pseudosphere provides a $C^\infty$-smooth isometric immersion of a horodisk in the hyperbolic plane into $E^3$. In particular, hyperbolic disks of arbitrarily large radius can be isometrically immersed into $E^3$.

4. It appears to be an open problem: What is the largest radius of a hyperbolic disk that can be $C^2$-smoothly isometrically embedded into $E^3$.

5. As a special case of Nash-Kuiper isometric embedding theorem, there exists a $C^1$-smooth isometric embedding of the hyperbolic plane into $E^3$.

6. For further reading:

Qing Han and Jia-Xing Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, AMS Mathematical Surveys and Monographs, vol. 130, 2006.