Chern’s conjecture on discrete constant scalar curvatures

Determine whether the possible constant scalar curvatures of closed minimal hypersurfaces in spheres form a discrete set; in the widely used refined formulation, establish whether every such hypersurface with constant scalar curvature is isoparametric.

Background

For a minimal immersion into the unit sphere, the Gauss equation gives Scal⁡M=n(n−1)−S\operatorname{Scal}_M=n(n-1)-S, so constant scalar curvature is equivalent to constant squared norm SS of the second fundamental form. The paper places its curvature-gap theorem in the context of the classical Chern program but explicitly states that the theorem does not resolve the hypersurface second-gap problem or the broader Chern conjecture.

The conjecture concerns the discreteness of constant scalar-curvature values for closed minimal hypersurfaces. A commonly used stronger formulation predicts that every closed minimal hypersurface with constant scalar curvature is isoparametric.

References

In the hypersurface case, Chern's conjecture is commonly formulated as the discreteness of the possible constant scalar curvatures; a widely used refined form predicts that every closed minimal hypersurface with constant scalar curvature is isoparametric, as discussed for example by Ge and Tang, Xu and Xu and Xu and Xu.

— A Curvature Gap for Minimal Submanifolds in Spheres  (2608.16095 - Li, 17 Aug 2026) in Section 1, Introduction

Thus, while Chern's conjecture fails for $m \geq 3$, its validity in codimension up to two when $n \geq 3$ remains an important open question.

— On Chern's conjecture for minimal submanifolds of the sphere  (2608.18074 - Firester et al., 18 Aug 2026) in Introduction, paragraph immediately following Theorem 1

Apart from Chang's work in dimension three and classification results under additional hypotheses, the discreteness problem remains open.

— Chern's Conjecture with Constant Cubic Trace  (2609.03711 - Tan et al., 3 Sep 2026) in Section 1, Introduction

The stronger classification expectation that every closed minimal hypersurface with constant scalar curvature is isoparametric remains open in general.

— Chern's Conjecture with Constant Cubic Trace  (2609.03711 - Tan et al., 3 Sep 2026) in Section 1, Introduction

The discreteness assertion in Question \ref{Chern-question} has traditionally been referred to as the Chern conjecture. Let $Mn$ be a closed minimal submanifold with constant scalar curvature in the $(n+q)$-dimensional unit sphere $\mathbb{S}{n+q}$. Then for each $n$ and $q$, the set of all possible values for $S$ is discrete.

— The pinching constant for closed minimal submanifolds of high codimension in the sphere  (2609.04631 - Xu et al., 4 Sep 2026) in Section 1, immediately following Question 1 / Chern Conjecture

In 1968 and 1970, Chern proposed the following famous conjecture. Let $M$ be a closed minimal submanifold with constant scalar curvature in the unit sphere $\mathbb{S}{n + 1}$. Then the set of all possible values of the scalar curvature of $M$ is discrete.

— Rigidity of Minimal Submanifolds in Spheres of Higher Codimension  (2609.04393 - Lei, 3 Sep 2026) in Section 1, Introduction, paragraph beginning “In 1968 and 1970” (Chern’s Conjecture)