Closed Minimal Hypersurfaces in $\mathbb{S}^5(1)$ with Constant Scalar and Gauss-Kronecker Curvatures

Abstract: In this paper, we prove that any closed minimal hypersurface $M<sup>4$ of $\mathbb{S}<sup>5(1)$ with constant scalar curvature and constant Gauss-Kronecker curvature must be isoparametric. Specifically, $M<sup>4$ is either an equatorial 4-sphere, a Clifford torus $$\mathbb{S}^{2\left(\frac{\sqrt{2}}{2}\right)\times} \mathbb{S}<sup>2\left(\frac{\sqrt{2}}{2}\right)$$ or $$\mathbb{S}^{1\left(\frac{1}{2}\right)\times} \mathbb{S}<sup>3\left(\frac{\sqrt{3}}{2}\right)$$, or a Cartan's minimal hypersurface. Consequently, the squared norm of the second fundamental form $S$ can only take the values 0, 4, 12. This result provides strong support for Chern's Conjecture.

• The paper proves that closed minimal hypersurfaces in S⁵(1) with constant scalar and Gauss-Kronecker curvatures are isoparametric, yielding only S=0, 4, or 12 for the squared norm of the second fundamental form.
• It employs weighted global differential forms and cut-off arguments to rigorously handle mixed multiplicity structures and delicate singular regions.
• The classification confirms the strong form of Chern’s conjecture in dimension 4, ruling out non-isoparametric cases and extending prior results in lower dimensions.

Classification of Closed Minimal Hypersurfaces in $\mathbb{S}^5(1)$ with Constant Scalar and Gauss-Kronecker Curvatures

Introduction and Mathematical Context

The paper addresses the longstanding classification problem for closed minimal hypersurfaces $M^4$ in the five-dimensional unit sphere $\mathbb{S}^5(1)$, subject to the condition of constant scalar curvature and constant Gauss-Kronecker curvature. This is situated within the broader framework of global differential geometry and the study of rigidity phenomena under curvature constraints. The main result is a complete classification, demonstrating that such hypersurfaces are necessarily isoparametric. Specifically, any closed minimal hypersurface in $\mathbb{S}^5(1)$ satisfying these curvature conditions is either totally geodesic (an equatorial 4-sphere), a Clifford torus (with two possible types), or Cartan's minimal hypersurface. It follows that the squared norm $S$ of the second fundamental form can only attain the values 0, 4, or 12, directly supporting strong forms of Chern's conjecture for $n=4$.

Background: Chern's Conjecture and Isoparametric Hypersurfaces

A central open question in this area, posed by S.S. Chern, is whether the set of possible values for $S$ is discrete for closed minimal hypersurfaces in spheres with constant scalar curvature. A stronger form conjectures that such hypersurfaces must be isoparametric; that is, their principal curvatures and associated multiplicities are constant. The landscape of results for arbitrary dimensions and codimensions is highly nontrivial, with significant progress often reliant on additional geometric or algebraic assumptions such as the constancy of higher-order power sums of the principal curvatures.

Isoparametric hypersurfaces in spheres are classified according to the number $g$ of distinct principal curvatures, which by Münzner's theorem can be only 1, 2, 3, 4, or 6. Each such value leads to exceptionally rigid geometry and a discrete spectrum for $S$.

Prior to this work, for $n=3$ the classification problem was resolved: in $M^4$0 only the equator, Clifford torus, and Cartan's minimal example arise under the hypotheses. For $M^4$1, previous results obtained the classification under stronger analytic or algebraic constraints (e.g., constant cubic form $M^4$2, Willmore hypotheses, or vanishing Gauss-Kronecker curvature), but not merely under the constancy of scalar and Gauss-Kronecker curvatures.

Main Results and Proof Strategy

The main theorem unequivocally asserts that any closed minimal hypersurface $M^4$3 with constant scalar curvature $M^4$4 and constant Gauss-Kronecker curvature $M^4$5 is isoparametric. Consequently, $M^4$6 is congruent (up to ambient isometry) to one of the following:

• An equatorial 4-sphere ($M^4$7)
• A Clifford torus, either $M^4$8 or $M^4$9 ($\mathbb{S}^5(1)$0)
• Cartan's minimal hypersurface in $\mathbb{S}^5(1)$1 ($\mathbb{S}^5(1)$2)

Strong supporting evidence is therefore provided for the strong form of Chern’s conjecture in dimension 4, as the only permissible values for $\mathbb{S}^5(1)$3 are discrete, and isoparametricity is forced under the stated curvature assumptions.

The proof is structured as a detailed case analysis based on the possible multiplicities and arrangements of the principal curvatures of the hypersurface:

1. Two distinct principal curvatures at a point: An explicit algebraic argument shows $\mathbb{S}^5(1)$4, and the hypersurface must be a Clifford torus (with both types realized).
2. Everywhere four distinct principal curvatures: Algebraic and global analytic arguments, leveraging integral formulas involving symmetric functions of the principal curvatures and global 3-form techniques, show $\mathbb{S}^5(1)$5 and force Cartan's minimal example. Key technical steps demonstrate that Gauss-Kronecker curvature must be positive and that scalar curvature is nonnegative (i.e., $\mathbb{S}^5(1)$6), with only $\mathbb{S}^5(1)$7 possible in this case.
3. Exactly three distinct principal curvatures at a point: This scenario is rigorously excluded via localized cut-off argument techniques, smoothing of minimum functions, and contradiction, thereby precluding any further exceptional cases.

No assumption of nonnegative scalar curvature or constant number of principal curvatures is required—these conditions are shown to be emergent from the initial hypotheses.

Methodological Innovations

The analytic core of the argument is founded on the construction and refined manipulation of globally defined weighted 3-forms, integration by parts, and local-to-global rigidity principles for symmetric functions of the principal curvatures. These 3-forms are intricately tailored to detect and exploit sign information in the relevant differential identities, such as the Laplacian of the fourth symmetric function $\mathbb{S}^5(1)$8, and encode rigidity phenomena through integral inequalities.

The proof overcomes several significant obstacles:

• Allowing mixed multiplicity structures: Careful local analysis and smoothing arguments handle transition regions where the configuration of principal curvatures changes.
• Avoiding a priori restriction on the sign of $\mathbb{S}^5(1)$9: The analytic method derives nonnegativity (and then the maximal value) as a necessary consequence, not a starting axiom.
• Handling delicate borderline singularities: Smoothing via mollification (as in Guan's lemma) provides control near singular strata of the stratification by eigenvalue multiplicity.

Implications and Connections

Theoretical Implications

This classification result confirms that the rigidity of closed minimal hypersurfaces in $\mathbb{S}^5(1)$0 under these curvature constraints is absolute—no exotic or non-isoparametric geometries arise. It removes the necessity for supplementary algebraic restrictions (such as constant higher power sums of principal curvatures) that were previously indispensable.

Moreover, this work demonstrates the potency of global differential forms and integration techniques for rigidity questions, suggesting applicability in higher codimensions and for other classes of submanifolds (e.g., with constant higher Gauss maps, higher mean curvatures, or Weingarten-type constraints).

Numerical Outcomes and Contradictory Claims

• Only the values $\mathbb{S}^5(1)$1 are admissible; in the context of the strong Chern conjecture, this is a sharp and exclusive result.
• All cases with three distinct principal curvatures are precluded, contradicting any expectation of further exceptional or non-isoparametric examples under these curvature assumptions.

Practical and Future Developments

For the geometric analysis community, this theorem closes the classification in dimension 4, matching the pattern seen in dimension 3. For higher dimensions ($\mathbb{S}^5(1)$2), and potentially in higher codimensions, the robustness of this analytic and global approach suggests promising avenues for analogous rigidity problems.

Potential further developments include:

• Extension to higher-dimensional spheres and classification under similar curvature prescriptions.
• Analysis of complete (not necessarily closed) minimal hypersurfaces with analogous constraints (relating to, for instance, Bryant's conjecture and its higher-dimensional analogs).
• Exploration of implications for spectral geometry, given the connection between isoparametric hypersurfaces and eigenfunctions of the Laplacian.

Recent independent works (cf. (Ge et al., 28 Jun 2026, He et al., 1 Mar 2026)) have obtained consistent classification theorems using different but related global techniques, confirming the correctness and broad relevance of the result.

Conclusion

The paper achieves a definitive classification of closed minimal hypersurfaces in $\mathbb{S}^5(1)$3 with constant scalar and Gauss-Kronecker curvatures, confirming that the only such hypersurfaces are isoparametric (equator, Clifford tori, Cartan example). This establishes the strong form of Chern's conjecture in dimension four under the natural geometric constraint of constant Gauss-Kronecker curvature. The integrated algebraic and analytic approach, especially the use of weighted differential forms and cut-off arguments, paves the way for further advances in the rigidity and classification theory for submanifolds in space forms.