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<p>Cheng-Hu-Moruz (2017) completely classified the locally strongly convex centroaffine hypersurfaces with parallel cubic form based on the Calabi product (called the type I Calabi product for short) proposed by Li-Wang (1991).</p><p>In the present paper, the authors introduce the type II Calabi product (in case λ<sub>1</sub> = 2λ<sub>2</sub>), complementing the type I Calabi product (in case λ<sub>1</sub> ≠ 2λ<sub>2</sub>), and achieve a classification of the locally strongly convex centroaffine hypersurfaces in ℝ<sup><i>n</i>+1</sup> with vanishing centroaffine shape operator and Weyl curvature tensor by virtue of the types I and II Calabi product.</p><p>As a corollary, 3-dimensional complete locally strongly convex centroaffine hypersurfaces with vanishing centroaffine shape operator are completely classified, which positively answers the centroaffine Bernstein problems III and V by Li-Li-Simon (2004).</p>

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Classification of the Conformally Flat Centroaffine Hypersurfaces with Vanishing Centroaffine Shape Operator

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • Miaoxin Lei,
  • Ruiwei Xu,
  • Peibiao Zhao

摘要

Cheng-Hu-Moruz (2017) completely classified the locally strongly convex centroaffine hypersurfaces with parallel cubic form based on the Calabi product (called the type I Calabi product for short) proposed by Li-Wang (1991).

In the present paper, the authors introduce the type II Calabi product (in case λ1 = 2λ2), complementing the type I Calabi product (in case λ1 ≠ 2λ2), and achieve a classification of the locally strongly convex centroaffine hypersurfaces in ℝn+1 with vanishing centroaffine shape operator and Weyl curvature tensor by virtue of the types I and II Calabi product.

As a corollary, 3-dimensional complete locally strongly convex centroaffine hypersurfaces with vanishing centroaffine shape operator are completely classified, which positively answers the centroaffine Bernstein problems III and V by Li-Li-Simon (2004).