# Curvatures of contours of solutions of 3d Poisson's equation

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 equation can be interpreted as the wave equation for a monochromatic wave, where $f$ is one component of electrical field. Surface of the same |f| are surfaces of the same intensity.

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Crossposted to mathoverflow.net/questions/87649/… (as in a week there were no comments or answers). –  Piotr Migdal Feb 6 '12 at 10:27