First of all, I must clarify at the outset that I am simply asking if there is an *alternative* way to solve an already known problem. It is known that the answer to my question is yes. The problem is as follows:

Consider $(\mathbb{R}^2, g_{H})$ with the hyperbolic metric, i.e.
$$ ds^2 = e^{2y} dx^2 + dy^2 .$$
Does there exist a smooth isometric immersion into $(\mathbb{R}^5, g_{flat})$? In other words we are looking for a smooth map $ u : \mathbb{R}^2 \rightarrow \mathbb{R}^5$ such that
$$ du_1(x,y)^2 + du_2(x,y)^2 + \ldots du_5(x,y)^2 = e^{2y} dx^2 + dy^2.$$
My question is the following: Is it conceivable that one can solve this question using the *continuity method*? More precisely, consider the family of metrics $g_t$ on $\mathbb{R}^2$ given by
$$ ds^2 = e^{2ty} dx^2 + dy^2 .$$
Clearly one can solve his pde when $t=0$. It would seem to me naively that the set of $t$ for which the pde is solvable is open (probably one can show this from the Implicit Function theorem).
Is it conceivable one can show this set is closed? Typically that is the difficult part of using the continuity method.

**Remark:** 1) There is an explicit solution to this immersion question. This can be found in the book "Isometric Embedding of Riemannian Manifolds in Euclidean Spaces" by Qing Han and Jia Xing Hong. The desired function $u$ is given explicitly.

2) Just in case the answer to my question is yes, i.e. one can actually use the continuity method to solve the immersion problem in $\mathbb{R}^5$, what are the obstructions to applying that same idea in $\mathbb{R}^4$? I believe it is an open question whether the hyperbolic plane can be immersed in $\mathbb{R}^4$.