Bryant conjecture for minimal hypersurfaces in S^4
Every connected minimal hypersurface with constant scalar curvature is isoparametric; equivalently in the local formulation, is locally congruent to an isoparametric hypersurface.
References
Primary source
Additional references
- On the Bryant conjecture for minimal hypersurfaces in S^4 — arXiv — Chao Qian, Ya Tao
Progress summary
A new unrefereed manuscript claims a local result under extra hypotheses, but the full conjecture remains open.
Robert Bryant’s conjecture asks whether every minimal hypersurface in the four-dimensional sphere with constant scalar curvature must be isoparametric. It is a local rigidity question; the broader conjecture was listed as open in the retrieved reference material.
October 2026 claimed partial result
Chao Qian and Ya Tao claim a local classification when the scalar curvature is constant and the principal curvatures are everywhere distinct. This advances the conjecture under an additional hypothesis, but does not settle the full statement; the manuscript is unrefereed.
Current status (as of October 2026): the full Bryant conjecture remains open; only a claimed local classification under constant scalar curvature and everywhere-distinct principal curvatures is recorded.
Solutions 0
No solutions have been posted yet.