{
  "abstract": "Let $M^n\\to\\mathbb{S}^{n+q}(1)$ be a closed connected minimal immersion, where $n\\ge3$, and set $Q=S+λ_2$, with $S=|h|^2$ and $λ_2$ the second largest eigenvalue of Lu's fundamental matrix. We determine the sharp codimension range for Lu's second-gap conjecture. For every $2\\le q\\le n$, there exists $γ_{n,q}>0$ such that, if $Q$ is constant and $Q>n$, then $Q\\ge n+γ_{n,q}$. Conversely, for every $q\\ge n+1$, we construct closed connected homogeneous minimal embeddings, followed when necessary by totally geodesic inclusions, with constant scalar curvature and constant $Q$-values dense in $(n,2n)$. Thus, in every dimension $n\\ge3$, Lu's conjecture holds precisely for $q\\le n$. Combined with the theorem of Peng-Terng for hypersurfaces and the recent resolution of the two-dimensional case, this gives a complete resolution of Lu's second-gap conjecture: for every $n\\ge2$, the conjecture holds exactly when $q\\le n$ and fails when $q\\ge n+1$. In codimension two we further obtain the explicit admissible gap $γ_{n,2}=\\exp(-10^{16}n^2)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33552",
  "author_total": 4,
  "authors": [
    {
      "name": "Weiran Ding"
    },
    {
      "name": "Fagui Li"
    },
    {
      "name": "Xize Yang"
    },
    {
      "name": "Yunheng Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7qnnmtc8mz01t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33552",
  "fulltext_url": null,
  "id": "7qnnmtc8mz01t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33552",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214794447",
  "pdf_url": "https://arxiv.org/pdf/2609.33552",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7qnnmtc8mz01t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2609.33552",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33552"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A solution to Lu's second gap conjecture",
  "title_text": "A solution to Lu's second gap conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/7qnnmtc8mz01t",
  "venue": "arXiv",
  "versions": []
}