关键词: 超曲面的 Ｃａｌａｂｉ几何, Ｈｅｓｓｉａｎ流形, 体积变分公式, 稳定极值曲面

Abstract: This paper firstly investigates the geometric structure of parametrized hypersurface under the Calabi normalization. Then, it is proved that the Calabi geometry of the general parametrized hypersurface is locally equivalent to the canonical Calabi normalization of the graphs of the convex functions. It is also proved that Hessian manifolds can be locally expressed as the typical Calabi geometry of the graphs with the convex functions. For parametrized hypersurface, the first volume variational formula and the second variational formula of the Calabi geometry are established. As a consequence, it is proved that any extreme Calabi surface with non-positive 2-dimensional Gauss curvature is stable, and the affine area functional obtains local maximum on such surfaces.