The sharp Chern--Simons--Sobolev inequality and its variants

We introduce and prove a new conformally invariant Chern–Simons–Sobolev inequality, and classify all optimizers, which are the Hopf map $\pi :{\mathbb{S}}^{2n+1}\to {\mathbb{C}\mathbb{P}}^n$ composed with a symplectic diffeomorphism of ${\mathbb{C}\mathbb{P}}^n$ and an orientation-preserving conformal transformation of ${\mathbb{S}}^{2n+1}$. The proof uses the sharp curl-Sobolev inequality and a rigidity argument for its equality case. Motivated by the main result, we also introduce the spherical Radu–Tchrakian–Yang functional, and prove the global minimality of the Hopf map for radii in a neighborhood of $((n!{)}^2\mathrm{/}n{)}^{1\mathrm{/}(4n-4)}$, and characterize all minimizers. We also prove local instability for sufficiently small and sufficiently large radii, with explicit lower bounds for the Morse index.