Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems
Flow by Gauss curvature to the Aleksandrov and dual Minkowski problems
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DOI：
10.4171/jems/936
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发表时间：
2017-12
影响因子：
2.6
通讯作者：
Qi-Rui Li;Weimin Sheng;Xu-jia Wang
中科院分区：
文献类型：
--
作者：
Qi-Rui Li;Weimin Sheng;Xu-jia Wang
本文研究欧氏空间R^{n+1}$中闭凸超曲面的收缩流，其速度为$fr ^{\alpha} K$，其中$K$是高斯曲率，$r$是超曲面到原点的距离，$f$是正光滑函数.如果$\alpha \ge n+1$，我们证明该流始终存在，并且在归一化为孤立子后平滑收敛，如果$f \等于1$，孤立子是以原点为中心的球体。我们的论证为经典的Aleksandrov问题提供了一个光滑范畴内的抛物性证明，并解决了Huang，Lutwak，Yang和Zhang提出的对偶q-Minkowski问题（Acta Math.216（2016）：325-388），对于q 0的情形，我们对偶函数f和原点对称的初始条件也建立了相同的结果，但是对于非对称的f，给出了上述光滑收敛性的一个反例。
In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^{\alpha} K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $\alpha \ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by Huang, Lutwak, Yang and Zhang (Acta Math. 216 (2016): 325-388), for the case $q 0$, we also establish the same results for even function $f$ and origin-symmetric initial condition, but for non-symmetric $f$, counterexample is given for the above smooth convergence.