Quermassintegral preserving local curvature flows

在一类包含单连通空间形式的柱形翘曲环境流形中，考虑了闭图超曲面和平均凸超曲面的逆曲率流.例如，通过将径向距离和角度函数的适当组合添加到反平均曲率流动方程中，可以得到减小总平均曲率的表面积保持流动。与以往的查询积分保持流不同，这种流的特点是它是局部的，不包含非局部强迫项。此功能具有几个技术优势。本课题的目的是证明这些流的长期存在性，并且对于图解和均值凸初始超曲面，光滑地收敛到一个坐标切片。作为应用，对于非凸超曲面，将遵循经典Alexandrov-Fichel不等式的几个推广。闭平均凸超曲面的一个Heintze-Karcher型不等式已被证明是沿着这类流动推导单调量的一个有用的工具。当且仅当超曲面是坐标切片时，该等式精确成立。这一结果的稳定性版本在研究这种曲率流的渐近性时将非常有用，因此也应在本项目中进行推导。

We consider inverse curvature flows with forcing terms of closed graphical and mean-convex hypersurfaces in a class of cylindrical warped ambient manifolds, which contains the simply connected spaceforms. For example, by adding suitable combinations of the radial distance and angle functions to the inverse mean curvature flow equation, one obtains a surface area preserving flow which decreases the total mean curvature. The special feature of this flow is that, contrary to previous quermassintegral preserving flows, it is local and does not contain a nonlocal forcing term. This feature has several technical benefits. The aim of this project is to deduce the long-time existence of these flows and smooth convergence to a coordinate slice for graphical and mean-convex initial hypersurfaces. As applications several generalisations of classical Alexandrov-Fenchel inequalities would follow for non-convex hypersurfaces. A Heintze-Karcher type inequality for closed mean-convex hypersurfaces has proven to be a helpful tool in the deduction of monotone quantities along such flows. This inequality holds with precise equality if and only if the hypersurface is a coordinate slice. A stability version of this result would be very useful in the investigation of the asymptotics of such curvature flows and hence shall also be deduced within this project.