A solution to Lu's second gap conjecture

Abstract

Let Mⁿ→𝕊^n+q(1) be a closed connected minimal immersion, where n≥3, and set Q=S+λ₂, with S=|h|² and λ₂ the second largest eigenvalue of Lu's fundamental matrix. We determine the sharp codimension range for Lu's second-gap conjecture. For every 2≤ q≤ n, there exists γ_n,q>0 such that, if Q is constant and Q>n, then Q≥ n+γ_n,q. Conversely, for every q≥ 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≥3, Lu's conjecture holds precisely for q≤ 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≥2, the conjecture holds exactly when q≤ n and fails when q≥ n+1. In codimension two we further obtain the explicit admissible gap γ_n,2=exp(−10¹⁶n²).

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BibTeX

@misc{ding2026solution,
  title = {{A solution to Lu's second gap conjecture}},
  author = {Weiran Ding and Fagui Li and Xize Yang and Yunheng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.33552},
  eprint = {2609.33552},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.33552},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - A solution to Lu's second gap conjecture
AU  - Weiran Ding
AU  - Fagui Li
AU  - Xize Yang
AU  - Yunheng Zhang
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2609.33552
UR  - https://arxiv.org/abs/2609.33552
AB  - Let M^n→𝕊^n+q(1) be a closed connected minimal immersion, where n≥3, 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≤ q≤ n, there exists γ_n,q>0 such that, if Q is constant and Q>n, then Q≥ n+γ_n,q. Conversely, for every q≥ 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≥3, Lu's conjecture holds precisely for q≤ 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≥2, the conjecture holds exactly when q≤ n and fails when q≥ n+1. In codimension two we further obtain the explicit admissible gap γ_n,2=exp(−10^16n^2).
ER  -

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