Sharp first-gap estimate at the Li–Li threshold

Establish that every closed non-totally geodesic minimal submanifold M^n of the unit sphere S^{n+q}, with n≥3 and q≥2, satisfies the sharp estimate S_max≥n.

Background

The main theorem proves the quantitative lower bound S_max>2n/3+(n−2)/182 for non-totally geodesic closed minimal submanifolds in codimension at least two. The paper observes that Clifford minimal hypersurfaces, regarded as submanifolds of higher-dimensional spheres through totally geodesic inclusions, have S≡n.

This motivates the explicitly stated conjecture that the optimal first-gap bound should be S_max≥n, equivalently that no non-totally geodesic example can have its maximum squared second fundamental-form length in the interval below n above the Li–Li threshold 2n/3.

References

However, for the cases of $n\geq 3$, it is still unknown whether there exists a pinching condition forcing $M$ to be a minimal submanifold with nonflat normal bundle.

— An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices  (2609.26309 - Liu et al., 22 Sep 2026) in Introduction, discussion following Theorem 2 (Lu’s pinching theorem)

Inspired by the main theorems of the present paper, we raise the following conjecture:

— An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices  (2609.26309 - Liu et al., 22 Sep 2026) in Introduction, immediately after the discussion of Lu’s conjecture and its known partial results

It is therefore natural to conjecture that every closed non-totally geodesic minimal submanifold $Mn\subset\mathbb{S}{n+q}$ with $n\geqslant 3$ and $q\geqslant 2$ satisfies the sharp estimate

S_{\max}\geqslant \frac{2n}{3}+\frac{n}{3}=n.

— The pinching constant for closed minimal submanifolds of high codimension in the sphere  (2609.04631 - Xu et al., 4 Sep 2026) in Section 1, paragraph immediately following the discussion of the preprint counterexamples