For every n,m≥ 1, the set of possible constant values of S=| h|^2 for closed, minimally immersed n-dimensional submanifolds M^n hookrightarrow S^n+m whose second fundamental form h has constant length is a discrete subset of R. Equivalently, the corresponding possible scalar curvatures form a discrete set.

Chern–do Carmo–Kobayashi conjecture

For every $n,m\ge 1$ , the set of possible constant values of $S=\lvert h\rvert^2$ for closed, minimally immersed $n$ -dimensional submanifolds $M^n\hookrightarrow \mathbb{S}^{n+m}$ whose second fundamental form $h$ has constant length is a discrete subset of $\mathbb{R}$ . Equivalently, the corresponding possible scalar curvatures form a discrete set.

Progress summary

Solved

A 2026 preprint gives broad families that violate the conjecture, so it is false in those dimension and codimension ranges.

The Chern–do Carmo–Kobayashi conjecture predicts that, for fixed dimension and codimension, the possible constant values of $S=\lvert h\rvert^{2}$ are discrete. The latest work refutes this assertion in broad ranges rather than proving the conjectured rigidity.

August 2026 counterexample

J. H. Eschenburg and J. H. S. de Lira report that the conjectured discreteness fails in broad dimension and codimension ranges, disproving Chern’s conjecture there. This is a direct counterexample claim in an arXiv preprint; related flat-torus examples for Lu’s refined quantity do not by themselves refute the original conjecture.

Current status (as of August 2026): The conjecture is refuted in the broad ranges covered by Eschenburg–de Lira, while any cases outside those ranges remain open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

On Chern's conjecture for minimal submanifolds of the sphere — arXiv — J. H. Eschenburg, J. H. S. de Lira

differential-geometry minimal-submanifolds

2 days ago