How to solve for the extrinsic variables of a one variable scaled conformal metric to an equivalent metric?

Question

Given the following metric equivalence \begin{align} e^{2w(x_2)} \left( dx_1^2+dx_2^2 \right) = dy_1^2+dy_2^2+dy_3^2 \end{align} is their a known solution for the extrinsic variables $y_1(x_1,x_2)$, $y_2(x_1,x_2)$ and $y_3(x_1,x_2)$? The scaling factor $e^{2w(x_2)}$ is only a function of $x_2$, not a function of both $x_1$ and $x_2$.

I know that the complex plane can be mapped to the surface of the Riemann sphere using the following metric \begin{align} \dfrac{4}{x_1^2+x_2^2+1} \left( dx_1^2+dx_2^2 \right) = dy_1^2+dy_2^2+dy_3^2 \end{align} where $x_1$ and $x_2$ are the complex variables of the complex plane and \begin{align} y_3 = \dfrac{x_1^2+x_2^2-1}{x_1^2+x_2^2+1} \\ y_2 = \dfrac{2x_2}{x_1^2+x_2^2+1} \\ y_1 = \dfrac{2x_1}{x_1^2+x_2^2+1} \end{align} is the transformation to the extrinsic variables of the Riemann sphere $y_1^2+y_2^2+y_3^2=1$.

Answer 1

^{tentative answer}

Your question can be phrased as follows: given a metric of the form $e^{2w(x_2)} \left( dx_1^2+dx_2^2 \right)$ on the plane, how to find an isometric immersion of this metric into $\mathbb R^3$? (An isometric immersion is a map $y=f(x)$ such that the relation in your post holds.) One can also go further and ask for isometric embedding. Either way, this is a difficult problem the study of which easily fills a book. E.g., it is still unknown whether every surface with $C^\infty$ smooth metric admits locally an isometric embedding into $\mathbb R^3$. Radially symmetric metrics $\exp(2w(x_1^2+x_2^2)) \left( dx_1^2+dx_2^2 \right)$ are easier to deal with, because one can look for a rotationally invariant surface in $\mathbb R^3$, thus reducing the problem from a difficult nonlinear PDE to an ODE. This works okay for metrics of positive curvature, but fails for negatively curved surfaces like the hyperbolic plane. (One can embed a piece of hyperbolic plane into $\mathbb R^3$ as a saddle-shaped surface, but this surface will not be rotationally symmetric despite the original metric being such.)

In your case, we have a different group of isometries acting on the surface: translations in the $x_1$ direction. Unfortunately we can't expect the image in $\mathbb R^3$ to also have translational invariance along $y_1$, because any such surface is of zero Gaussian curvature.

So, I expect that there is no explicit way to find a local embedding in general, and that a global embedding need not exist at all. One thing I would try:

1. Apply the exponential map $z=e^{x_2+ix_1}$, which is a local isometry of the given surface onto the plane with the radially symmetric metric $\frac{\exp(2 w(\log |z|))}{|z|^2}|dz|^2$
2. Use a radial stretch map in the plane to turn the metric into the form $dr^2+f(r)^2\,d\theta^2$.
3. Follow the ODE-based method as in this paper.

For a computational approach to the embedding problem, see undergraduate paper by Sonya Kim.