Beltrami fields with a nonconstant proportionality factor are rare

We consider the existence of Beltrami fields with a nonconstant proportionality factor $f$ in an open subset $U$ of $\mathbf{R}^3$. By reformulating this problem as a constrained evolution equation on a surface, we find an explicit differential equation that $f$ must satisfy whenever there is a nontrivial Beltrami field with this factor. This ensures that there are no nontrivial solutions for an open and dense set of factors $f$ in the $C^k$ topology. In particular, there are no nontrivial Beltrami fields whenever $f$ has a regular level set diffeomorphic to the sphere. This provides an explanation of the helical flow paradox of Morgulis, Yudovich and Zaslavsky.Comment: 13 page

Laplace operators with eigenfunctions whose nodal set is a knot

We prove that, given any knot $\gamma$ in a compact 3-manifold M, there exists a Riemannian metric on M such that there is a complex-valued eigenfunction u of the Laplacian, corresponding to the first nontrivial eigenvalue, whose nodal set $u^{-1}(0)$ has a connected component given by $\gamma$. Higher dimensional analogs of this result will also be considered.Comment: 16 page