Year: 2026
Type: preprint
Abstract: We introduce and prove a new conformally invariant Chern--Simons--Sobolev inequality, and classify all optimizers, which are the Hopf map $π:\mathbb{S}^{2n+1}\to\mathbb{CP}^n$ composed with a symplectic diffeomorphism of $\mathbb{CP}^n$ 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 $((n!)^2/n)^{1/(4n-4)}$, and characterize all minimizers. We also prove local instability for sufficiently small and sufficiently large radii, with explicit lower bounds for the Morse index.
Source: arXiv (Cornell University)
Author: Zhang, Mingwei
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223228319