Math Conjectures

Chern Conjecture for Minimal Hypersurfaces

OPENLandmarkConjectureProposed 1986 · Standard version

Canonical statement

Let Mn⊂Sn+1(1)M^n\subset S^{n+1}(1) be a closed minimally immersed hypersurface. If MM has constant scalar curvature, equivalently if the squared norm ∣A∣2|A|^2 of its second fundamental form is constant, then MM is isoparametric: all of its principal curvatures are constant.
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Let \(M^n\subset S^{n+1}(1)\) be a closed minimally immersed hypersurface. If \(M\) has constant scalar curvature, equivalently if the squared norm \(|A|^2\) of its second fundamental form is constant, then \(M\) is isoparametric: all of its principal curvatures are constant.
GEO-DIFF-025:[ChernDoCarmoKobayashi1970Minimal][Verstraelen1986SectionalCurvature][ScherfnerWeissYau2012Chern][FiresterTsiamis2026Chern]
Geometry and topologyPartial differential equations

Notes

For a closed minimal hypersurface Mn⊂Sn+1M^n\subset S^{n+1}, the Gauss equation gives ScalM=n(n−1)−∣A∣2\mathrm{Scal}_M=n(n-1)-|A|^2, so constant scalar curvature is equivalent to constant squared norm of the second fundamental form. The later strong Chern conjecture recorded here says that every such hypersurface is isoparametric, hence has constant principal curvatures; it grew from the rigidity results of Chern, do Carmo, and Kobayashi [ChernDoCarmoKobayashi1970Minimal] and the 1986 strong formulation [Verstraelen1986SectionalCurvature].

The conclusion is known in selected dimensions, pinching ranges, and cases with extra constancy assumptions, with isoparametric hypersurfaces supplying the model examples. Scherfner, Weiss, and Yau survey these rigidity and gap theorems and explain why proving that every constant-∣A∣2|A|^2 minimal hypersurface is isoparametric remains difficult [ScherfnerWeissYau2012Chern].

Firester and Tsiamis constructed counterexamples to a broader higher-codimension version of Chern's original discreteness expectation [FiresterTsiamis2026Chern]. Their submanifolds are not hypersurfaces, so they do not refute the strong codimension-one isoparametric statement recorded here; that problem remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.