Question

Hilbert proved that there's no complete regular (C^k for sufficiently large k) isometric embedding of the hyperbolic plane into R^3. On the other hand, the pseudosphere is locally isometric to the hyperbolic plane up to its cusps (though it has the topology of a cylinder). What's the largest hyperbolic disk (with Gaussian curvature -1) that can be smoothly (or C^2, say) isometrically embedded in R^3?

edit: This doesn't seem to be getting many views, so I'll bump this by adding in a rather easy lower bound from the pseudosphere. First, the pseudosphere is parametrized by the region PS={z | Im z ≥ 1, -π < Re z ≤ π} on the upper half plane model of H^2. Let z=x+iy, so that ordered pairs (x,y)∈ H^2 when y>0.

Next, Euclidean circles drawn on in the upper-half plane model with center (x,y\cosh r) and radius y\sinh r correspond to hyperbolic circles with center (x,y) and radius r. I can fit a Euclidean circle of radius π centered at (0,1+π) into the region PS. This corresponds to a hyperbolic disk of radius arctanh(π/(1+π)) ~ 0.993.

Surely one can do better?

edit2: fixed mistakes in formulas above (didn't affect the bound). Here're some pictures:

Answer 1

I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ² into ℝ³. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded horodisk, seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.

Answer 2

I think that you can get an arbitrarily large disk. The proof is by crochet. Since there's a pattern for crocheting constant negative curvature disks where you increase the radius as you go and since we live in 3-space, it follows that you can get arbitrarily large disks.

See this TED talk for some cool applications of hyperbolic crochet to biology, or this article for a more rigorous explanation.

Answer 3

I can't comment on Noah's answer, so:

The reason for the C^1 condition is that Nash's C¹ embedding theorem says that any Riemannian k-manifold with a short embedding into Rⁿ has an isometric C¹ embedding into Rⁿ for any n > k. In particular, there is an isometric C¹ embedding of the hyperbolic plane into R³. The very beautiful proof proceeds by "pleating" the embedding to be closer and closer to isometric, while keeping it contained in a not-much-larger region.

Answer 4

Several russian geometers have addressed this question. I suggest you a survey on Isometric immersions by A. Borisenko (2001, I think) in Russian Mathematical Surveys (it is in English)