Divergence-free framings of three-manifolds via eigenspinors

TL;DR

This paper offers a new proof, inspired by Seiberg-Witten theory, that any closed orientable three-manifold with a volume form admits three orthogonal, divergence-free vector fields of equal non-zero length, using eigenspinors.

Contribution

It provides an alternative, spinor-based proof of Gromov's theorem on divergence-free vector fields on three-manifolds, connecting geometric analysis with topological results.

Findings

Existence of three orthogonal divergence-free vector fields of equal length

New proof method using eigenspinors and Seiberg-Witten theory

Applicable to any Riemannian metric on the manifold

Abstract

Gromov used convex integration to prove that any closed orientable three-manifold equipped with a volume form admits three divergence-free vector fields which are linearly independent at every point. We provide an alternative proof of this (inspired by Seiberg-Witten theory) using geometric properties of eigenspinors in three dimensions. In fact, our proof shows that for any Riemannian metric, one can find three divergence-free vector fields such that at every point they are orthogonal and have the same non-zero length.