发表机构
Henan Normal University（河南师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过构造两类完全中心仿射极值超曲面，反驳了中心仿射Bernstein猜想，表明在欧氏完备性（$n\ge2$）和中心仿射完备性（$n\ge3$）下该猜想均不成立。
AI 中文摘要
关于双曲中心仿射极值超曲面的中心仿射Bernstein猜想预测，非负Ricci曲率结合欧氏完备性或中心仿射完备性会迫使超曲面属于Wang类。我们通过系统研究Calabi复合来反驳该猜想：我们发展了一种构造完全中心仿射极值超曲面的通用几何方法，并应用该方法构造了两个违反此刚性的族。第一族在欧氏度量下完备但中心仿射度量不完备，具有非负Ricci曲率，且在二维中具有正Gauss曲率；与一个点迭代复合可给出所有维度$n\ge2$的例子。第二族在三维中平坦且对两种度量均完备，迭代可给出所有维度$n\ge3$的例子。因此，中心仿射Bernstein猜想在欧氏完备性下对$n\ge2$失败，在中心仿射完备性下对$n\ge3$失败。
英文摘要
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 $n\ge2$. The second family is flat and complete in both metrics in dimension three, and iteration yields examples in every dimension $n\ge3$. Thus the centroaffine Bernstein conjecture fails under Euclidean completeness for $n\ge2$ and under centroaffine completeness for $n\ge3$.
Comments23 pages, Comments are welcome
