Beltrami Fields with Morse Proportionality Factor

Abstract

In this work we study Beltrami fields with non-constant proportionality factor on $\mathbb{R}^3$. More precisely, we analyze the existence of vector fields $X$ satisfying the equations $curl(X)=fX$ and $div(X)=0$ for a given $f\in C^\infty(\mathbb R^3)$ in a neighborhood of a point $p\in\mathbb{R}^3$. Since the regular case has been treated previously, we focus on the case where $p$ is a non-degenerate critical point of $f$. We prove that for a generic Morse function $f$, the only solution is the trivial one $X\equiv 0$ (here generic refers to explicit arithmetic properties of the eigenvalues of the Hessian of $f$ at $p$). Our results stem from the introduction of algebraic obstructions, which are discussed in detail throughout the paper.