3 problems

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Killing-form minimizer conjecture for the three-sphere curl quotient
Killing-form minimizer conjecture. Killing $1$ -forms, equivalently Killing vector fields, are global minimizers of the quotient defining $S_1$ . Consequently,
Conjecturerefutedcurl-sobolev-inequalities riemannian-geometry three-spherearXiv posted 2 days ago
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Frank–Loss conjecture on the extremizer for the curl-Sobolev constant
Frank–Loss conjecture. The infimum $S_3$ is achieved by $\xi$ .
Conjectureopencurl-sobolev-inequalities differential-geometry extremal-problemsarXiv posted 2 days ago
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Round-sphere optimizer conjecture for the first curl-Sobolev constant
Round-sphere optimizer conjecture. The standard round metric $g_{\rm st}$ on $\mathbb{S}^n$ is an optimizer for $\mu([g])$ .
Conjectureopenconformal-geometry curl-sobolev-inequalities spherearXiv posted 2 days ago

Killing-form minimizer conjecture for the three-sphere curl quotient