超曲面Calabi几何的体积变分及稳定性

Volume variational formulae for the Calabi geometryof hypersurface and the stability

首先给出了参数化超曲面在Calabi法化下的几何结构。证明了一般参数化超曲面的Calabi几何均可局部描述为凸函数的图的典型Calabi几何,并证明Hessian流形可局部表示为凸函数的图的典型Calabi几何。对于参数化超曲面,建立了Calabi几何的体积第一变分公式和第二变分公式。作为推论,证明了2维Gauss曲率非正的极值Calabi曲面是稳定的,并且仿射面积泛函在这类曲面取得极大值。

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.