几何不等式与几何刚性是整体微分几何中的重要研究课题，等周不等式和球面的刚性定理是超曲面几何中的两个典型代表。本项目将重点研究以下几个超曲面几何中的不等式与刚性问题：.1. 静态流形中超曲面的带权重的等周型Alexandrov-Fenchel型不等式与球中具有自由边界超曲面的相对Alexandrov-Fenchel不等式；.2. 球中具有自由边界或常接触角的极小曲面或常平均曲率曲面的刚性问题；.3. 各向异性Gauss曲率流的收敛性与各向异性Gauss孤立子的刚性问题。.在本项目的实施过程中，拟综合使用微分几何、积分几何以及几何分析中的各类技巧，包括几何变分方法、积分方法、极值原理等。

Geometric inequalities and geometric rigidity are central research subjects in global differential geometry. Typical examples in geometry of hypersurfaces are isoperimetric inequality and rigidity of sphere. This project is concerned with the following geometric inequalities and rigidity problems in geometry of hypersurfaces..1. Weighted isoperimetric type Alexandrove-Fenchel inequality for hypersurfaces in static manifolds; relative Alexandrove-Fenchel inequality for hypersurfaces with free boundary in a ball..2. Rigidity problems on minimal or constant mean curvature hypersurfaces with free boundary or constant contact angle in a ball..3. Convergence of anisotropic Gauss curvature flow and rigidity of anisotropic Gauss soliton..In this project, several techniques in differential geometry, integral geometry and geometric analysis will be utilized and developed, such as geometric variational method, integral method and maximum principle.