I think you are asking the wrong question.

By scaling you have that if you can solve for

$$ \mathrm{d}s^2 = e^{2ty} \mathrm{d}x^2 + \mathrm{d}y^2 $$

for any non-zero $t$ you can also solve for

$$ \mathrm{d}s^2 = e^{2sy} \mathrm{d}x^2 + \mathrm{d}y^2 $$

for any non-zero $s$.

Proof: Let $u'(x,y) = u(\lambda x,\lambda y)$, you have $|\mathrm{d}u'|^2 = e^{2\lambda y} \lambda^2(\mathrm{d} x^2 + \mathrm{d}y^2)$. So you just need to define $\tilde{u} = \lambda^{-1} u'$ to get the desired embedding.

This means that if the solution contains any point $t_0 \in \mathbb{R}\setminus\{0\}$ it will also contain the entirety of $\mathbb{R}\setminus\{0\}$.

So the continuity method doesn't do anything for you: regardless of whether the solution set is closed, what you really need to prove is that there exists some $t_0$ for which the differential equations are solvable. Or, in other words, the hard part lies in showing

an open neighborhood of $\{0\}$ lies in the solution set,

the part which you implicitly considered to be obvious.

Also, the question of immersion of hyperbolic plane in $E^4$ is partially known, at least if you allow piecewise smooth solutions. See

• http://www.ams.org/mathscinet-getitem?mr=1025303
• http://www.ams.org/mathscinet-getitem?mr=1334978

where immersions are constructed which are globally $C^{0,1}$ and real analytic away from some singular lines.

I think you are asking the wrong question.

By scaling you have that if you can solve for

$$ \mathrm{d}s^2 = e^{2ty} \mathrm{d}x^2 + \mathrm{d}y^2 $$

for any non-zero $t$ you can also solve for

$$ \mathrm{d}s^2 = e^{2\tilde{t}y} \mathrm{d}x^2 + \mathrm{d}y^2 $$

for any non-zero $\tilde{t}$.

Proof: Let $u'(x,y) = u(\lambda x,\lambda y)$, you have $|\mathrm{d}u'|^2 = e^{2\lambda y} \lambda^2(\mathrm{d} x^2 + \mathrm{d}y^2)$. So you just need to define $\tilde{u} = \lambda^{-1} u'$ to get the desired embedding.

This means that if the solution contains any point $t_0 \in \mathbb{R}\setminus\{0\}$ it will also contain the entirety of $\mathbb{R}\setminus\{0\}$.

So the continuity method doesn't do anything for you: regardless of whether the solution set is closed, what you really need to prove is that there exists some $t_0$ for which the differential equations are solvable. Or, in other words, the hard part lies in showing

an open neighborhood of $\{0\}$ lies in the solution set,

the part which you implicitly considered to be obvious.

Another way to say the above is: the method of continuity is helpful if you want to get to "1" or "$\infty$" knowing that you have "0". If you can already reduce the problem to getting to "$\epsilon$", the method of continuity is not going to tell you much.

Also, the question of immersion of hyperbolic plane in $E^4$ is partially known, at least if you allow piecewise smooth solutions. See

• http://www.ams.org/mathscinet-getitem?mr=1025303
• http://www.ams.org/mathscinet-getitem?mr=1334978

where immersions are constructed which are globally $C^{0,1}$ and real analytic away from some singular lines.