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?