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