Hyperbolic diameter of Amsler's surface

Question

I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of intersection for the two lines that generate it. But I guess I'm only interested in the most symmetric version, where these lines are orthogonal. I'd like to know how large a portion of the hyperbolic plane I can embed on this surface.

So if the unit of length is chosen such that the Gaussian curvature of the surface becomes $-1$, what is the radius of a circle which is centered at the intersection of the asymptotic lines and just touches the cuspidal edges of the surface? A “circle” here would be the set of points on the surface with fixed geodesic distance to a central point. That intrinsic circle would not be a planar circle in the 3D embedding of the surface.

The following illustration of Amsler's surface was taken from the Gallery of pseudospherical surfaces by A. Ovchinninkov. I'm not sure how accurately it matches what I ask for, since there are other figures on the web which look somewhat different from this.

Answer 1

this question was studied in http://arxiv.org/abs/1005.4442 . The figure above was taken from Amsler's original paper and the dashed curve closely matches the largest geodesic circle cut from the Amsler surface.

Interestingly, the authors show that arbitrarily large isometric embeddings of the hyperbolic plane exist. They show this by taking an Amsler surface with an angle of $\phi_n=\frac{\pi}{n}$ between the two generating lines and periodically extending the surface bounded between these two lines. However, it is still true that there does not exist an isometric embedding of the entire hyperbolic plane.

Answer 2

The previous answer points to the paper http://arxiv.org/abs/1005.4442, by Gemmer and Venkataramani [GV], which tells us that the radius requested is $1.2654$. But the previous answer makes some claims about the GV paper that might mislead.

The GV paper shows that hyperbolic disks of arbitrarily large radius can be isometrically and real-analytically immersed in 3-space. But the immersion described in [GV] is not injective for large disks, so it isn't an embedding. They immerse a large disk by letting it wrap many times around the pseudosphere.

This lack of injectivity can be easily fixed, however. Dini's surface is a twisted analog of the pseudosphere that is simply connected, rather than being cylindrical. So a hyperbolic disk of arbitrarily large radius can be isometrically and real-analytically embedded in 3-space as a disk-shaped subset of Dini's surface. Note that, with this technique, large disks will be wrapped, many times, very tightly around the screw axis of Dini's surface.

The previous answer also reports on results from [GV] about periodic Amsler surfaces. It is important to realize that the periodic Amsler surfaces defined in [GV] are only $C^1$ smooth; they fail to be $C^2$ along the joint where two pieces meet. This lack of smoothness is not a big issue for Gemmar and Venkataramani, who are actually trying to model how real-life thin disks will crumple when forced to assume negative curvature. It's not clear how interesting a $C^1$ embedding of a disk would be to a mathematician, however, since the entire hyperbolic plane can be isometrically embedded in 3-space, if we are willing to settle for just $C^1$ smoothness.

Answer 3

The actual value of the maximal radius turns out to be $1.2662$, just a skosh larger than the $1.2654$ that Gemmmer and Venkataramani give.

Amsler's 1955 paper is available at https://eudml.org/doc/160462, albeit in the original French. I wrote Mathematica code that solves the differential equations in that paper numerically, after which we can make 3D plots of Amsler's surface and its maximal circle. Here is a static image:
It is much more instructive, of course, to view the image interactively, using your mouse to explore various points of view. I encourage you to do this, using either Mathematica itself or the free Wolfram CDF Player. (Amsler himself might have enjoyed playing with these images; but a web query suggests, sadly, that he died five years ago.) The following link should enable you to download either my Mathematica notebook file "AmslerSurface.nb" or the corresponding CDF file "AmslerSurface.CDF" from a Google Drive directory of mine named "Amsler surface":

https://drive.google.com/open?id=0B6eoYeIB8EaqaENJbHZJcHdEa1E

In my notebook, I explain where the various differential equations come from and the little glitches that arose as I solved them with Mathematica. (If anyone has suggestions for how I could improve my Mathematica code, I'd be happy to hear.)

Evaluating all of the Mathematica commands in my notebook takes between 8 and 10 hours, the vast majority of which goes into the three commands that do the actual 3D rendering. You won't have to reevaluate those commands just to view the graphics interactively, with either Mathematica or CDF Player. But some people might want to do more with the images, such as composing them with newly created elements. To avoid forcing those people to wait for such a reevaluation, I also provide the outputs of those three rendering commands as separate Mathematica source files, with extension ".m", as explained at the end of the notebook "AmslerSurface.nb".

Answer 4

A comment on the answer by Lyle Ramshaw. The surfaces in http://arxiv.org/abs/1005.4442 Gemmer and Venkataramani[GV] are actually $C^{1,1}$ so only barely off the $C^2$ regularity hypothesis for the Efimov theorem on the non-existence of isometric immersions of complete hyperbolic surfaces with uniformly negative curvature. In particular, the surfaces in [GV] are much smoother than the $C^1$ immersions given by the Nash embedding.

The fact that we only need to ``break" $C^2$ regularity on a small set, where we can have $C^{1,1}$ regularity goes back to the work of Rozendorn in the 1960's (cf. Geometry III in the Encyclopedia of Mathematical Sciences series of Springer books)