Quermassintegral preserving local curvature flows

考虑了一类包含单连通空间形式的圆柱弯曲环境流形中闭图超曲面和平均凸超曲面的带强迫项的反曲率流。 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方程，获得减小总平均曲率的表面积保持流。这个流的特殊之处在于，与以前的quermassintegral保持流相反，它是局部的，并且不包含非局部强制项。此功能具有几个技术优势。这个项目的目的是推导出这些流的长期存在性和光滑收敛到一个坐标切片的图形和平均凸初始超曲面。作为应用，经典的Alexandrov-Fenchel不等式的几个推广将遵循非凸超曲面。关于闭平均凸超曲面的Heintze-Karcher型不等式已被证明是推导沿着这类流的单调量的有用工具。当且仅当超曲面是坐标切片时，这个不等式精确相等。这个结果的稳定性版本在研究这种曲率流的渐近性时将是非常有用的，因此也将在这个项目中推导出来。

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.