{"paper":{"title":"The sharp curl-Sobolev inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.DG","authors_text":"Guofang Wang, Mingwei Zhang","submitted_at":"2026-07-26T20:22:30Z","abstract_excerpt":"We solve a longstanding problem, going back at least to Rivi\\`ere 1998 and open even in the physically most relevant case $n=3$, by proving a sharp curl-Sobolev inequality on $\\mathbb{S}^n$ when $n\\equiv 3\\pmod 4$: for every $\\frac{n-1}{2}$-form $\\alpha$, the conformally invariant quotient satisfies (with positive denominator) \\[ \\frac{\\Big(\\int_{\\mathbb{S}^n}|{\\rm curl}\\alpha|^{\\frac{2n}{n+1}}\\,{\\rm dV}\\Big)^{\\frac{n+1}{n}}}{\\int_{\\mathbb{S}^n}\\langle{\\rm curl}\\alpha,\\alpha\\rangle\\,{\\rm dV}}\n  \\ge \\frac{n+1}{2}\\,\\omega_n^{\\frac1n}. \\] We also classify all extremals in terms of Killing forms. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.23827","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.23827/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}