Let $f(x,y,z)$ be a complex function in a 3d euclidian space that fulfill the Poisson's equation $$\frac{\partial^2}{\partial x^2} f + \frac{\partial^2}{\partial y^2} f + \frac{\partial^2}{\partial z^2} f = c,$$ where c is a real number.

Let $X$ be a surface being a contour surface of $|f|^2$, i.e. $|f(x,y,z)|^2=k$ for some $k\geq0$.

Can anything be said on the curvature (mean or gaussian) of $X$? (I.e. if it is bounded from above or below? If, within one surface, its changes are bounded?)

Motivation: I am interested in the penetration length of evanescent waves. Then the Poisson's equation can be interpreted as the wave equation for a monochromatic wave, where $f$ is one component of the electric field. Surface of the same $|f|^2$ are surfaces of the same intensity.