Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up
Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up
复制标题
DOI：
10.1515/crelle-2017-0019
复制
发表时间：
2016-03
期刊：
影响因子：
--
通讯作者：
J. Isenberg;Haotian Wu
中科院分区：
文献类型：
--
作者：
J. Isenberg;Haotian Wu
摘要研究了旋转对称非紧嵌入超曲面的平均曲率流中的II型曲率爆破现象。使用分析技术的基础上正式匹配渐近和建设的上，下障碍解决方案包络正式解决方案与规定的行为，我们表明，对于每个初始超曲面考虑，平均曲率流解决方案表现出以下行为附近的“消失”时间T：（1）最大曲率集中在超曲面的顶点（脐点），并且对于参数γ > 1 2 {\gamma>\frac{1}{2}}的每个选择，有一个最大曲率的解以（T-t）-（γ + 1 2）{（T-t）^{{-（\gamma+\frac{1}{2}）}}的速率爆炸。(2)在尖端附近，解收敛到一个平移孤子，这是一个更高的维模拟的“死神”解决方案的曲线缩短流。(3)远离尖端，流面以取决于参数γ的特征速率接近一个坍缩圆柱。
Abstract We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T: (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter γ > 1 2 {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate ( T - t ) - ( γ + 1 2 ) {(T-t)^{{-(\gamma+\frac{1}{2})}}} . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ.