论文标题
riemannian歧管的等距曲面作为平均曲率流量的初始数据
Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow
论文作者
论文摘要
我们表明，只要平均弯曲流的独特性在初始数据和相应的环境空间中，平均曲率流的唯一性和相应的环境空间只要平行家族的独特性和相应的环境空间，平均曲率流的异位层曲面的演变是通过平均曲率流的歧管。作为一种应用，我们提供了一类Riemannian流形，这些歧管允许具有恒定主曲线的超曲面，这些曲线不是等术超曲面。此外，对于一类环境空间，我们表明，由于初始数据是I型奇异性，因此由平均曲率流带有等于曲面的平均曲率流形成的奇异性。 We apply our results to describe the evolution of isoparametric hypersurfaces by the mean curvature flow in ambient spaces with nonconstant sectional curvature, such as homogenous 3-manifolds $\mathbb{E}(κ, τ)$ with 4-dimensional isometry groups, and Riemannian products $\mathbb{Q}^2_{c_1} \times \ Mathbb {q}^2_ {C_2} $的空间表单。
We show that the evolution of isoparametric hypersurfaces of Riemannian manifolds by the mean curvature flow is given by a reparametrization of the parallel family in short time, as long as the uniqueness of the mean curvature flow holds for the initial data and the corresponding ambient space. As an application, we provide a class of Riemannian manifolds that admit hypersurfaces with constant principal curvatures, which are not isoparametric hypersurfaces. Furthermore, for a class of ambient spaces, we show that the singularities developed by the mean curvature flow with isoparametric hypersurfaces as the initial data are Type I singularities. We apply our results to describe the evolution of isoparametric hypersurfaces by the mean curvature flow in ambient spaces with nonconstant sectional curvature, such as homogenous 3-manifolds $\mathbb{E}(κ, τ)$ with 4-dimensional isometry groups, and Riemannian products $\mathbb{Q}^2_{c_1} \times \mathbb{Q}^2_{c_2}$ of space forms.
