Let a surface be given by $z = f(x,y)$. Assume that, for some $a>0$, $f:[-a,a]\times[-a,a]\to\mathbb{R}$ and that $k_i(x,y)$ are the principal curvatures of the surface.
Can we find the surface from the principal curvatures $k_i$?
We can write a parameterization $\sigma(x,y)=(x, y, f(x,y))$, so the question would equivalent to determining $f$ in terms of the principal curvatures $k_i$. I think one could consider normal sections on the surface, for which we can compute the curvature using the formula \begin{equation} \kappa(t)=\frac{f''(t)}{(1+f'(t)^2)^{3/2}}, \end{equation} since these curves are graphs of $f$, but I'm not sure if the whole argument works fine. And even if it does, we would need to solve non-linear PDEs.
Is there a simpler way of finding the surface?