For each pair of dimensions n,m, there exists varepsilon_n,m>0 such that every compact minimal submanifold M^n⊂ S^n+m satisfying M cong S^n and n≤∑_alpha=1^minn,mlambda_alpha+lambda_2<n+varepsilon_n,m is a Veronese manifold, where lambda_alpha are the eigenvalues of the fundamental matrices of the second fundamental form.

Eigenvalue pinching conjecture for compact minimal submanifolds in spheres

The conjecture predicts that sufficiently near the lower eigenvalue-pinching threshold, a compact minimal submanifold homeomorphic or diffeomorphic to a sphere must be a Veronese manifold. No proposer or date is identified in the retrieved sources.

Known results

Z. Q. Lu: the condition $|B|^2+\lambda_2\leq n$ forces total geodesicity, a generalized Clifford torus, or a Veronese surface.

2026 preprint: optimal all-codimension pinching theorem

Huimin Liu and Ling Yang claim that, for a compact minimal $n$ -manifold in $S^{n+m}$ , the bound $$\sum_{\alpha=1}^{\min{n,m}}\lambda_\alpha+\lambda_2\leq n $forces total geodesicity or equality. They state that equality is optimal, with generalized Clifford tori and Veronese manifolds attaining it, and give the expected classifications in dimensions$ 2 $and$ 3$. This advances the conjecture by removing the codimension restriction but does not establish the broader near-threshold statement; the preprint is unverified.

Current status (as of September 2026): A claimed sharp pinching theorem now covers arbitrary codimension, while the stated near-threshold Veronese characterization remains unproved and the claim lacks independent verification.

Eigenvalue pinching conjecture for compact minimal submanifolds in spheres

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