On Chern's Conjecture for Minimal Submanifolds with Flat Normal Bundle in Spheres

Abstract: Let $M<sup>n$ $(n\geqslant3)$ be a closed minimal submanifold in the unit sphere $\mathbb S<sup>{n+m}$ $(m\geqslant2)$ with flat normal bundle, and let $S$ denote the squared norm of its second fundamental form. We prove an explicit second-gap rigidity theorem for $S$. More precisely, if $S$ is constant and [ 0\leqslant S\leqslant n+δ, ] where $δ$ is an explicit constant satisfying $δ\geqslant \frac{n}{87}$, then either $S\equiv0$ and $M$ is a totally geodesic sphere, or $S\equiv n$ and $M$ is a Clifford torus contained in a totally geodesic $$\mathbb S^{{n+1}\subset\mathbb} S<sup>{n+m}$$. %We observe that the flat-normal-bundle assumption is necessary here. The flat-normal-bundle condition is essential in the general higher-codimensional setting: without it, the corresponding rigidity statement already fails in dimension two. This theorem provides positive evidence for Chern's conjecture in higher codimension.

• The paper establishes an explicit second-gap rigidity theorem, proving that closed minimal submanifolds with flat normal bundle in the unit sphere must either be totally geodesic or a Clifford torus when the squared norm of the second fundamental form is within a specified range.
• It leverages advanced extensions of Simons' formula and Peng–Terng-type invariants, combining analytic and algebraic methods to derive sharp numerical gap constants dependent on both dimension and codimension.
• The work underscores the crucial role of the flat normal bundle in achieving rigidity, offering refined numerical thresholds that extend classical rigidity results and support the discrete nature of key curvature values posited by Chern's conjecture.

Explicit Second-Gap Rigidity for Minimal Submanifolds with Flat Normal Bundle in Spheres

Introduction and Context

The paper "On Chern's Conjecture for Minimal Submanifolds with Flat Normal Bundle in Spheres" (2607.10733) addresses longstanding questions in submanifold geometry, specifically focusing on the Chern conjecture for closed minimal submanifolds in unit spheres $\mathbb{S}^{n+m}$ with flat normal bundle. The conjecture posits that the possible values of the squared norm of the second fundamental form, $S$, for such minimal submanifolds (with constant scalar curvature) are discrete, generalizing rigidity phenomena first observed by Simons, Chern, Peng–Terng, and others.

Previous work had largely resolved the constant-$S$ rigidity in codimension one, but explicit gap results in higher codimension, even under strong geometric assumptions like normal bundle flatness, were missing. This paper establishes such a second-gap rigidity theorem for all $n\geqslant 3$ and all codimension $m\geqslant 2$, providing both sharp estimates and demonstrating the essential role of flatness in the normal bundle.

Main Results

Let $M^n$ be a closed minimal submanifold of the unit sphere $\mathbb{S}^{n+m}$ ($n \geqslant 3$, $m \geqslant 2$) with flat normal bundle, and let $S$ denote the squared norm of its second fundamental form. The paper proves the following:

1. Explicit Second-Gap Rigidity: If $S$0 is constant and satisfies $S$1, where $S$2 is an explicit constant (at least $S$3 for $S$4), then $S$5 falls into one of two categories:
   • $S$6 and $S$7 is a totally geodesic sphere.
   • $S$8 and $S$9 is a Clifford torus lying in a totally geodesic $S$0.
2. Sharpness and Necessity of Flatness: The flat normal bundle assumption is essential; without it, even in $S$1 (surface case), rigidity fails—explicit constructions by Li–Zhao show sequences of flat minimal tori (non-Clifford) in spheres with $S$2 and normal curvature constants tending to zero.
3. Codimension Dependence: The explicit gap constants $S$3 depend on the dimension and codimension, with full numerical values derived for all relevant cases.

Technical Methods and Innovations

The proof synthesizes advanced analytic and algebraic techniques:

• Simons' Formula and Extensions: Higher-codimensional generalizations of Simons' formula are employed, with careful calculation of the Laplacians of $S$4 and $S$5, taking advantage of the flatness in the normal bundle to diagonalize the second fundamental form.
• Peng–Terng-Type Invariants: A new higher-codimensional invariant, $S$6, is introduced and bounded using sharp algebraic inequalities derived via the moving frame method.
• Gap Estimates via Integral and Pointwise Inequalities: Both integral and pointwise inequalities are established for $S$7, $S$8, and associated quartic invariants. The gap constants are optimized using Young’s inequality, sharp Cauchy–Schwarz variants, and intricate numerical optimization for the required estimates.
• Codimension \textit{m}=2 Specialization: For codimension two, the gap constants improve due to refined estimates, with full calculations showing the parameters where all analytic inequalities yield the desired rigidity.
• Numerical Verification: All gap constants are underpinned by precise numerical verification, ensuring thresholds are met in the full range of dimensions considered.

Strong Numerical and Rigidity Results

• For $S$9, $n\geqslant 3$0, the explicit second-gap constant $n\geqslant 3$1 ensures no minimal submanifolds (with flat normal bundle and constant $n\geqslant 3$2 in $n\geqslant 3$3) except the Clifford torus.
• In codimension two, sharper constants are obtained ($n\geqslant 3$4 for $n\geqslant 3$5, and $n\geqslant 3$6 for $n\geqslant 3$7), via more delicate estimates.
• All gap constants, invariants, and estimates are carefully optimized and their sharpness verified, with full numerical calculations presented in detailed appendices.

Implications and Future Directions

Theoretical Implications

This work provides the first explicit second-gap rigidity theorem for the constant-$n\geqslant 3$8 Chern-type problem in genuinely higher codimension, thus fundamentally extending classical rigidity results for minimal submanifolds. It substantiates the expectation that the set of possible $n\geqslant 3$9 values is discrete in the class of minimal submanifolds with flat normal bundle, opening avenues for further exploration of rigidity phenomena in geometric analysis and the classification problem for minimal submanifolds in spheres.

The results also clarify the limitations of normal scalar curvature alone as a rigidity criterion, emphasizing the necessity of the flat normal bundle condition in higher codimension.

Practical Applications

While focused within pure geometric analysis, the methods employed—particularly the sharp algebraic inequalities and techniques for controlling quartic invariants—may transfer to other contexts in differential geometry and potentially to geometric problems arising in theoretical physics (e.g., in the study of calibrated geometries and string theory, where minimality and curvature conditions are crucial).

Speculation on Future Developments

• Further refinements of gap constants and rigidity phenomena may be possible in special cases, or under weaker assumptions (e.g., partial flatness).
• A deeper understanding of the interplay between normal bundle curvature and gap rigidity may shed light on the structure of moduli spaces of minimal submanifolds in spheres.
• The algebraic techniques for bounding invariants may find applications in higher-order curvature flows and in the analysis of singularities for geometric PDEs.

Conclusion

The paper (2607.10733) decisively advances the theory of rigidity for minimal submanifolds in spheres with flat normal bundle, proving explicit second-gap theorems for constant $m\geqslant 2$0 and establishing the necessity of the flatness assumption. The results not only verify the discrete nature of $m\geqslant 2$1 values (as posited by Chern’s conjecture) in higher codimension, but also provide strong analytic and algebraic tools for further exploration in the field. The nuanced synthesis of analytic and algebraic methods, combined with rigorous numerical verification, sets a new benchmark for research on minimal rigidity in geometric analysis.