{"paper":{"title":"Flat minimal tori and Lu's second-gap conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DG","authors_text":"Fagui Li, Yuhang Zhao","submitted_at":"2026-06-29T15:11:01Z","abstract_excerpt":"Lu's first pinching theorem states that a closed minimal $n$-dimensional submanifold of the unit sphere satisfying $0\\le S+\\lambda_2\\le n$ is one of the standard first-gap models; here $S$ is the squared norm of the second fundamental form and $\\lambda_2$ is the second eigenvalue of Lu's fundamental matrix. Lu's second-gap conjecture asserts that, once $S+\\lambda_2$ is constant and strictly larger than $n$, it is separated from $n$ by a positive gap depending only on the dimension and codimension. We construct closed embedded counterexamples for minimal surfaces in every codimension at least t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.30432","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/2606.30432/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"}