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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 Ο€:π•Š2n+1β†’β„‚β„™n composed with a symplectic diffeomorphism of β„‚β„™n and an orientation-preserving conformal transformation of π•Š2n+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/(4nβˆ’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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Cite

BibTeXRIS
BibTeX
@misc{zhang2026sharp,
  title = {{The sharp Chern--Simons--Sobolev inequality and its variants}},
  author = {Mingwei Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29769},
  eprint = {2610.29769},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29769},
}
RIS
TY  - UNPB
TI  - The sharp Chern--Simons--Sobolev inequality and its variants
AU  - Mingwei Zhang
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29769
UR  - https://arxiv.org/abs/2610.29769
AB  - We introduce and prove a new conformally invariant Chern--Simons--Sobolev inequality, and classify all optimizers, which are the Hopf map Ο€:π•Š^2n+1β†’β„‚β„™^n composed with a symplectic diffeomorphism of β„‚β„™^n and an orientation-preserving conformal transformation of π•Š^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/(4nβˆ’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.
ER  -