4 problems

Consider the Nicole energy … and the Aratyn–Ferreira–Zimerman energy … for maps $u:\mathbb{R}^3\to\mathbb{S}^2$. A Vakulenko–Kapitansky-type bound means a lower bound of the form…

For smooth finite-energy fields $u:\mathbb{R}^3\to\mathbb{S}^2$, let … and let $Q(u)$ be the Hopf invariant. The Vakulenko–Kapitansky inequality is…

Rivière's conjecture. The Hopf map $\pi$ is a global minimizer of this $3$-energy in its homotopy class. The conjecture was supported by evidence among symmetric fibrations, while…

Let $\mathcal{FS}_\rho$ be the perturbed Faddeev–Skyrme energy on maps $u:\mathbb S^3\to\mathbb S^2$, and let $h:\mathbb S^3\to\mathbb S^2$ be the Hopf map. Two maps are in the sam…