# Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?

Consider the following question:

"Can the hyperbolic plane $(\mathbb{R}^2, g_H)$ be isometrically immersed in three dimensional Eulidean space$(\mathbb{R}^3, g_{flat})$?"

I believe the answer to this question is no. Can someone give me a reference for this theorem (in particular I want to look at the details of the proof and understand why this is not possible).

My second question is as follows: I assume the answer is no if one is asking for smooth (i.e. $\mathcal{C}^{\infty}$) immersion. Is anything known if we relax this condition? More precisely, asking for an immersion is asking whether a certain pde has a solution. What happens if I am just looking for some "weak" solution to this pde?

• This is known as Hilbert's theorem en.wikipedia.org/wiki/… . If I recall correctly there is a discussion of this as well as other "no-go" theorems in the book "Isometric Embedding of Riemannian Manifolds in Euclidean Spaces" by Qing Han and Jia Xing Hong which you have cited in another question. See also this question mathoverflow.net/questions/111101/… – j.c. Aug 20 '13 at 16:39
• Thank you for the wikepedia reference. I do have the book you mentioned; I am not certain however if he discusses any notion of a "weak" solution for an immersion there. – Ritwik Aug 20 '13 at 16:43

The Nash-Kuiper embedding theorem applies here, as the obvious smooth topological embedding of the hyperbolic plane as the unit disk in 3-dimensional Euclidean space decreases distances. Therefore there is a $C^1$ isometric embedding. I don't know of a stronger result.
Klotz-Milnor proved in 1972 that there is no $C^2$ isometric embedding of $H^2$ in $R^3$ so Nash-Kuiper can not be improved http://www.zentralblatt-math.org/zmath/scans.html?volume_=236&count_=348
• Note that the theorem is really due to Efimov (who proved that there is complete, $C^2$ surface in $R^3$ with curvature $K\leq -\epsilon<0$ -- this applies in particular to the hyperbolic plane). The paper by Klotz-Milnor is a (very nice) exposition of Efimov's result. – Jean-Marc Schlenker Aug 27 '13 at 19:22