论文标题
在双曲线空间中保留凸出突出的高斯曲率流量
Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space
论文作者
论文摘要
我们考虑双曲线空间中平滑，闭合和凸出曲面的体积流量$ \ mathbb {h}^{n+1}（n \ geq 2）$，其速度由高斯曲率的任意正力$α$给出的速度。我们证明，如果初始超出表面是凸，则流的平滑解仍然凸，并且在[0，\ infty）$中的所有正时间$ t \中存在。此外，我们采用了Kohlmann的结果，该结果使用双曲线曲率测量量和Alexandrov反射的论点来表明该流量在平滑拓扑中呈指数呈指数呈指数收敛。这可以看作是非本地类型体积保留曲率流的第一个结果，用于双曲线空间中的高空曲面，而初始数据仅需要凸度。
We consider the volume preserving flow of smooth, closed and convex hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1} (n\geq 2)$ with the speed given by arbitrary positive power $α$ of the Gauss curvature. We prove that if the initial hypersurface is convex, then the smooth solution of the flow remains convex and exists for all positive time $t\in [0,\infty)$. Moreover, we apply a result of Kohlmann which characterises the geodesic ball using the hyperbolic curvature measures and an argument of Alexandrov reflection to prove that the flow converges to a geodesic sphere exponentially in the smooth topology. This can be viewed as the first result for non-local type volume preserving curvature flows for hypersurfaces in the hyperbolic space with only convexity required on the initial data.
