I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help.

Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold and consider solving the following pde near a surface $\Sigma$  $<d\Phi,d\Phi>_g =0$ where $\Phi$ is a complex valued function.

If the dimenion is 2 then any function of $z=x_1 + i x_2$ would do the job in isothermal coordinates.

I understand the question is a bit vague but I am interested in general observations about solvability of this pde. Thanks,

I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help.

Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold and consider solving the following pde near a surface $\Sigma$ 

$<d\Phi,d\Phi>_g =0$ where $\Phi$ is a complex valued function.

If the dimenion is 2 then any function of $z=x_1 + i x_2$ would do the job in isothermal coordinates.

I understand the question is a bit vague but I am interested in general observations about solvability of this pde. Thanks,