Complete centroaffine extremal hypersurfaces and centroaffine Bernstein conjecture
Abstract
The centroaffine Bernstein conjecture for hyperbolic centroaffine extremal hypersurfaces predicts that nonnegative Ricci curvature together with either Euclidean or centroaffine completeness forces the hypersurface to lie in Wang's class. We disprove it by systematically studying Calabi compositions: we develop a general geometric method for constructing complete centroaffine extremal hypersurfaces and apply it to construct two families that violate this rigidity. The first family is Euclidean complete but has incomplete centroaffine metric, nonnegative Ricci curvature, and positive Gauss curvature in dimension two; iterating the composition with a point gives examples in every dimension n≥2. The second family is flat and complete in both metrics in dimension three, and iteration yields examples in every dimension n≥3. Thus the centroaffine Bernstein conjecture fails under Euclidean completeness for n≥2 and under centroaffine completeness for n≥3.
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BibTeX
@misc{xing2026complete,
title = {{Complete centroaffine extremal hypersurfaces and centroaffine Bernstein conjecture}},
author = {Cheng Xing and Yalin Sun and Ruiwei Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09332},
eprint = {2610.09332},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09332},
}
RIS
TY - UNPB
TI - Complete centroaffine extremal hypersurfaces and centroaffine Bernstein conjecture
AU - Cheng Xing
AU - Yalin Sun
AU - Ruiwei Xu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09332
UR - https://arxiv.org/abs/2610.09332
AB - The centroaffine Bernstein conjecture for hyperbolic centroaffine extremal hypersurfaces predicts that nonnegative Ricci curvature together with either Euclidean or centroaffine completeness forces the hypersurface to lie in Wang's class. We disprove it by systematically studying Calabi compositions: we develop a general geometric method for constructing complete centroaffine extremal hypersurfaces and apply it to construct two families that violate this rigidity. The first family is Euclidean complete but has incomplete centroaffine metric, nonnegative Ricci curvature, and positive Gauss curvature in dimension two; iterating the composition with a point gives examples in every dimension n≥2. The second family is flat and complete in both metrics in dimension three, and iteration yields examples in every dimension n≥3. Thus the centroaffine Bernstein conjecture fails under Euclidean completeness for n≥2 and under centroaffine completeness for n≥3.
ER -