A Gap Theorem for the Möbius Cross Energy Near the Hopf Link

We prove that the critical value $2{\pi }^2$ is isolated for the Möbius cross energy of two-component links in the round three dimensional sphere ${\mathbb{S}}^3$. Specifically, there exists ${\epsilon }_0>0$ such that any non-split, regular $H^2$ pair of curves with disjoint images, having vanishing first variation and energy at most $2{\pi }^2+{\epsilon }_0$, must lie in the Möbius-reparametrization orbit of the standard Hopf link.