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arXiv · 2610.29769

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

Mingwei Zhang
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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.

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Mingwei Zhang. 2026-10-21. The sharp Chern--Simons--Sobolev inequality and its variants. https://arxiv.org/abs/2610.29769

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