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

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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.

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## 1 Answer

There's nothing you can say locally, other than that the level surfaces of $|f|^2$ are real-analytic (when they are surfaces). Conversely, any embedded real-analytic surface $S$ in $\mathbb{R}^3$ has an open neighborhood $U$ on which there is a function $f:U\to\mathbb{C}$ that satisfies $\Delta f = c$ on $U$ and $|f|^2 = \lambda$ on $S$, where $\lambda>0$ is an arbitrarily specified constant. Moreover, you can arrange that the differential of $|f|^2$ along $S$ is nonzero, so that it is, effectively isolated.

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