5 problems

Let $M$ be a closed oriented $3$-manifold with volume form $\mu$, and let $X$ be an exact divergence-free vector field with $\omega=\iota_X\mu=d\alpha$. Its helicity is … A functio…

Let $X$ be a vector field in the setting of three-dimensional volume-preserving flows, and let its corresponding helicity level set be the set of vector fields with the same helici…

Nontrivial-helicity conjecture. If $\Sigma$ has non-trivial homology, then there exists $v\in C^\infty\mathcal{V}_0(\Sigma)$ with $\mathcal{H}(v)\neq 0$.

Helicity isoperimetric nonexistence conjecture. The corresponding helicity isoperimetric problem does not admit a solution.

Let $\Omega$ be a domain with $1$-dimensional homology, and let $(0,2,3)$-helicity be the helicity invariant associated with this domain. Nontriviality conjecture. If $\Omega$ has…