紧致型对称空间中的等焦超曲面与反向平均曲率流
Equifocal hypersurfaces in symmetric spaces of compact type and backward mean curvature flows
AI总结：
研究紧致型单连通不可约对称空间中等焦超曲面，推导其平均曲率和形状算子平方范数公式，通过分析反向平均曲率流沿长时间解的平均曲率等的时间演化，推广相关结果并扩展猜想。
AI中文摘要：
我们首先推导了紧致型单连通不可约对称空间中等焦超曲面的平均曲率和形状算子平方范数的公式，这些公式根据等焦超曲面的切向焦点数据明确给出。接着，我们研究了等焦超曲面的反向平均曲率流。Liu和Radeschi证明了等焦超曲面该流的长时间存在性。我们分析了沿长时间解的平均曲率和形状算子平方范数的时间演化，推广了Liu和Terng关于球面上等参超曲面的结果，还将Liu - Terng关于球面上反向平均曲率流的猜想扩展到紧致型单连通不可约对称空间。
英文摘要：
We first derive a formula for the mean curvature and the squared norm of the shape operator of equifocal hypersurfaces in simply-connected irreducible symmetric spaces of compact type. The formulas are given explicitly in terms of the tangential focal data of the equifocal hypersurfaces. Third, we study the backward mean curvature flow for equifocal hypersurfaces. The long-time existence of this flow for an equifocal hypersurface was established by Liu and Radeschi. We analyze the time evolution of the mean curvature and the squared norm of the shape operator along the long-time solution, thereby we generalize the result of Liu and Terng for isoparametric hypersurfaces in the sphere. Our analysis also gives an extension of Liu-Terng conjecture on the backward mean curvature flows in the sphere to the simply-connected irreducible symmetric space of compact type.
