Is there an analogue of the fundamental theorem of surfaces for complex surfaces?


If I know only differentiable functions $E,F,G,e,f,g$ (coefficients of the first and second fundamental forms) where $e,f,g$ are holomorphic functions,
can I use the equation of Gauss and Codazzi?
If not, is there an analogue for complex surfaces?



Analogue of fundamental theorem of real surfaces for complex surfaces
For any complex-valued real analytic functions $E,F,G,e,f,g$ of real variables $u,v$ satisfying the Gauss--Codazzi equations, there is a real analytic immersion of the set of points where $EG-F^2 \ne 0$ to a complex surface inside 3-dimensional complex Euclidean space inducing the first fundamental form $E \, du^2 + 2F \, du \, dv + G \, dv^2$ with second fundamental form $e \, du^2 + 2f \, du \, dv + g \, dv^2$. The immersion is unique up to complex affine transformation with linear part in the complex orthogonal group. The proof is identical to the usual proof, after complexification of the functions, as the standard proof only uses the Frobenius theorem, which has the obvious generalization to the complex analytic category. I doubt if this immersion will be of any use to you.


