For each integer n≥ 2, let V_n=c∈R:there exists a closed embedded minimal hypersurface M^n⊂S^n+1(1) whose shape operator A satisfies |A|^2 equiv c. Chern's conjecture asserts that V_n is a discrete subset of R; equivalently, every c∈V_n is isolated among the possible constant values of |A|^2.

Chern’s conjecture

The conjecture asks whether the possible constant values of the squared curvature $|A|^2$ of closed minimal hypersurfaces in spheres form a discrete set. It remains unresolved in full generality.

Known results

Ding–Xin (2011) proved rigidity under $0\le S-n\le n/23$ .
Xu–Xu (2016) improved this pinching range to $0\le S-n\le n/22$ .
Wei–Xu (2007), Zhang, Peng–Terng, and Cheng–Ishikawa established corresponding low-dimensional cases.
Under constant traces through order $n-1$ , a 2020 result forces the next trace to be constant and the hypersurface to be isoparametric; a 2021 result gives further bounds when $\operatorname{tr}(A^3)$ is constant.

September 3, 2026 local-finiteness theorem

A paper claims that, assuming $\operatorname{tr}(A^3)$ is constant, the value set of $|A|^2$ is locally finite without fixing topology or the cubic-trace value. This is genuine progress in a constrained setting, but it does not settle the unrestricted conjecture and remains unverified.

Current status (as of September 2026): The general discreteness conjecture remains open; local finiteness is claimed only under constant $\operatorname{tr}(A^3)$ , and that claim has not been independently verified.

Chern’s conjecture

Primary source

Additional references

Progress summary

Solutions 0