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…

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…

Ward's conjecture. The coefficient $3^{3/8}16\pi^2$ in this bound can be significantly improved. The exponent $3/4$ is known to be optimal, so the conjecture concerns only the coef…