论文标题
球形帽的平均曲率刚度和非rigigity导致
Mean Curvature Rigidity and Non-rigidity Results on Spherical Caps
论文作者
论文摘要
我们证明，欧几里得空间中的半球$ r^{n+1} $，被视为函数的图形，不承认光滑的扰动，因为具有平均曲率$ h \ ge 1 $的图形，其边界赤道固定为$ c^2 $。这是Gromov和Souam在空间形式中完全脐带超出表面的\ emph {平均曲率刚度}现象的\ emph {平均曲率刚度}现象的扩展。证明使用切线原则。另一方面，我们表明，在一个出色的球形盖上存在$ h \ ge 1 $的非平凡平滑扰动，其边界的边界固定为$ c^2 $。类似的结果对于减少$ h $的扰动和$ r $平均曲率函数$ H_R $的相似结果。在圆圈和离散对象的一维情况下，刚性和非依据之间的这种对比甚至是正确的（刻在圆圈中的多边形）。
We prove that a hemisphere in the Euclidean space $R^{n+1}$, viewed as the graph of a function, admits no smooth perturbations as graphs with mean curvature $H\ge 1$ whose boundary equator is fixed up to $C^2$. This is an extension of the \emph{Mean Curvature Rigidity} phenomenon discovered by Gromov and Souam on non-compact totally umbilic hypersurfaces in space forms. The proof uses a Tangency Principle. On the other hand, we show that there exist nontrivial smooth perturbations with $H\ge 1$ on a great spherical cap whose boundary is fixed up to $C^2$. Similar results hold true for perturbations decreasing $H$, and for the $r$ mean curvature function $H_r$. This contrast between rigidity and non-rigidity is even true in the 1-dimensional case for circles and for discrete objects (polygons inscribed in a circle).
