TITLE: Can the hyperbolic plane be immersed in three dimensional Euclidean space, if we are only looking for a weak solution?
QUESTION [12 upvotes]: 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?

REPLY [9 votes]: 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 
A practical realisation of an embedding is here: http://www.math.cornell.edu/~dwh/papers/crochet/crochet.html