3d Laplacian vector field with constant magnitude

Question

Are there non-trivial (i.e. non-constant) 3D Laplacian vector fields with constant magnitude? If not, how can one proof this, if yes, how many are there (how can they be classified?)

Answer 1

If $\Delta \vec F=0$ and $|\vec F|$ is constant, then $\vec F$ is constant. Indeed, let $u(x) = \vec F(0)\cdot \vec F(x)$. This is a harmonic function with $u(0)=|\vec F(0)|^2$ and $|u(x)|\le |\vec F(0)|^2$ for all $x$. By the maximum principle, $u$ is identically equal to $|\vec F(0)|^2$. This implies $\vec F(x)\equiv \vec F(0)$.