Hypersurfaces with flat centroaffine metric and equations of associativity

Classification of the locally strongly convex centroaffine hypersurfaces with parallel cubic form

In this paper, we study locally strongly convex centroaffine hypersurfaces with parallel cubic form with respect to the Levi-Civita connection of the centroaffine metric. As the main result, we obtain a complete classification of such centroaffine hypersurfaces. The result of this paper is a centroaffine version of the complete classification of locally strongly convex equiaffine hypersurfaces with parallel cubic form due to Hu, Li and Vrancken [12].

An optimal inequality on locally strongly convex centroaffine hypersurfaces

In this paper, we establish a general inequality for locally strongly convex centroaffine hypersurfaces in $\mathbb{R}^{n+1}$ involving the norm of the covariant derivatives of both the difference tensor $K$ and the Tchebychev vector field $T$. Our result is optimal in that, applying our recent classification for locally strongly convex centroaffine hypersurfaces with parallel cubic form in [4], we can completely classify the hypersurfaces which realize the equality case of the inequality.

Completeness of hyperbolic centroaffine hypersurfaces

This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity conditions on the boundary of the convex cone generated by the hypersurface. The main result is that completeness holds for hyperbolic components of level sets of homogeneous cubic polynomials. This implies that every such component defines a complete quaternionic Kähler manifold of negative scalar curvature.

Every centroaffine Tchebychev hyperovaloid is ellipsoid

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking observation that such hypersurfaces are characterized by having a Riemannian structure admitting a canonically defined closed conformal vector field. Then, by taking the advantage of properties about Riemannian manifolds with closed conformal vector fields, we show that the ellipsoids are the only centroaffine Tchebychev hyperovaloids. This solves the longstanding pr…

关于中心仿射超曲面分类的一个注记

A Note on the Classification of Centroaffine Hypersurfaces

本文指出Cao-Wang关于中心仿射超曲面的分类定理中，退化情形下的额外假设可以去掉，从而加强原定理。

完全中心仿射极值超曲面与中心仿射Bernstein猜想

Complete centroaffine extremal hypersurfaces and centroaffine Bernstein conjecture

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。