Quermassintegral preserving local curvature flows

考虑一类包含单连通空间形式的圆柱形弯曲环境流形中具有封闭图形和平均凸超曲面强迫项的逆曲率流。例如，通过将径向距离和角度函数的适当组合加入到逆平均曲率流动方程中，可以得到减小总平均曲率的保持表面积的流动。该流的特殊之处在于，与以往的保持quermass积分的流不同，它是局部的，不包含非局部强迫项。这个特性有几个技术上的好处。这个项目的目的是推断这些流的长期存在和平滑收敛到一个坐标切片的图形和平均凸初始超曲面。对于非凸超曲面，经典亚历山德罗夫-芬切尔不等式的几个推广将被应用。闭平均-凸超曲面的Heintze-Karcher型不等式已被证明是沿此类流推导单调量的有用工具。当且仅当超曲面是坐标切片时，此不等式成立。这一结果的稳定性版本对于研究此类曲率流的渐近性非常有用，因此也将在本项目中推导出来。

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.