论文标题
关于非全能线性HOPF流的收敛性
On the Convergence of Non-Integer Linear Hopf Flow
论文作者
论文摘要
考虑了曲率方程式的线性组合旋转对称表面的演变。众所周知，如果系数形成某些整数比率，则流量平滑，可以明确集成。在本文中，对于某些系数值以及初始表面上有轻度的分析限制，请考虑非全能情况。 我们证明，如果在初始表面上的北极和南极的焦点重合，则流量会收敛到圆形球体。否则，流量会收敛到非共同的HOPF球体。还给出了初始表面极点上散光的条件，以确保流动的收敛性。 该证明使用奇异sturm-liouville操作员的光谱理论来构建一个特征性，以在适当的空间中显示出演化会收敛的适当空间。
The evolution of a rotationally symmetric surface by a linear combination of its radii of curvature equation is considered. It is known that if the coefficients form certain integer ratios the flow is smooth and can be integrated explicitly. In this paper the non-integer case is considered for certain values of the coefficients and with mild analytic restrictions on the initial surface. We prove that if the focal points at the north and south poles on the initial surface coincide, the flow converges to a round sphere. Otherwise the flow converges to a non-round Hopf sphere. Conditions on the fall-off of the astigmatism at the poles of the initial surface are also given that ensure the convergence of the flow. The proof uses the spectral theory of singular Sturm-Liouville operators to construct an eigenbasis for an appropriate space in which the evolution is shown to converge.
