Centroaffine Bernstein conjecture
Let be a hyperbolic centroaffine extremal hypersurface. The centroaffine Bernstein conjecture asserts that if has nonnegative Ricci curvature and is complete either with respect to the Euclidean metric or with respect to its centroaffine metric, then belongs to Wang's class.
References
Primary source
Additional references
- Complete centroaffine extremal hypersurfaces and centroaffine Bernstein conjecture — arXiv — Cheng Xing, Yalin Sun, Ruiwei Xu
Progress summary
A 2026 preprint claims to disprove the conjecture by constructing complete counterexamples, but the claim has not been independently checked.
The conjecture comprises five centroaffine Bernstein problems posed by Li–Li–Simon in 2004, concerning rigidity of complete centroaffine extremal hypersurfaces. The latest work claims constructions that answer all five problems.
Known results
- Lei–Xu–Zhao had previously given positive answers to Problems III and V.
- The January 2026 preprint constructed examples addressing the remaining problems, including an elliptic hypersurface in and complete hyperbolic examples.
October 2026 claimed resolution
Cheng Xing, Yalin Sun, and Ruiwei Xu claim two Calabi-composition families: one Euclidean-complete and another complete for both relevant metrics. Their October 2026 preprint states that these constructions answer all five problems and thereby refute the broad rigidity assertion; this remains an unverified claim.
Current status (as of October 2026): A preprint claims a complete resolution through counterexamples, but no independent verification is recorded, so the problem remains open as a verified result.
Solutions 0
No solutions have been posted yet.