Singularities in crystalline curvature flows
Singularities in crystalline curvature flows
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晶体曲率流中的奇点
DOI：
10.4310/ajm.2002.v6.n1.a6
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发表时间：
2001
影响因子：
0.6
通讯作者：
B. Andrews
中科院分区：
文献类型：
--
作者：
B. Andrews
本文讨论了在晶体曲率流下平面内移动的多边形凸曲线的行为，其中每条边的运动速度由其长度的函数决定。该行为取决于边缘长度接近零时速度的增长率：对于缓慢增长（包括速度与长度的幂 α ε (0, 1) 成反比的均匀情况），总是存在封闭面积接近零而长度保持正值的解决方案。如果 α > 1，则所有解都渐近于拟似收缩解，如果 α = 1，则出现一系列不同类型的奇点。 1. 晶体曲率流 自从在[T]中引入以来，一些作者已经考虑了平面内多边形曲线的晶体曲率流。我们建议读者参考 [TCH] 和 [AG] 来讨论此类流动的几何和物理动机。出于目前的目的，我们仅考虑凸曲线，尽管可以更普遍地定义流量。在这种情况下，流可以通过以下方式定义：令 γ 为平面中的闭合凸 N 边多边形，并标记边 γ0,…。 。 。 , γN−1 按逆时针顺序排列。令 θi ∈ S = R/2πZ 为 γi 的外法线角度，并令 `i 为 γi 的长度。通过晶体曲率流移动 γ 包括找到从 γ 开始的连续族多边形曲线 γ(t)，以便每条边保持相同方向，但以由其长度确定的速度 vi 沿向外法线方向移动： (1) vi(t) = gi(`i)。这里 gi 是定义在 (0,∞) 上的平滑函数，对于每个 i 单调递增。本文主要关注收缩流，其中 gi 0 和 fi 对于每个 i 都是正实数。 1991年数学学科分类。 53C44、52A10、34C11。 1
This paper discusses the behaviour of polygonal convex curves in the plane moving under crystalline curvature flows, in which the speed of motion of each edge is determined by a function of its length. The behaviour depends on the rate of growth of the speed as the length of the edge approaches zero: For slow growth — including the homogeneous case where speed is inversely proportional to a power α ∈ (0, 1) of the length — there are always solutions for which the enclosed area approaches zero while the length remains positive. If α > 1, then all solutions are asymptotic to homothetically contracting solutions, and if α = 1 then there is a range of different kinds of singularity that occur. 1. Crystalline curvature flows Several authors have considered crystalline curvature flows of polygonal curves in the plane, since their introduction in [T]. We refer the reader to [TCH] and [AG] for a discussion of the geometric and physical motivation for such flows. For present purposes we consider only convex curves, although the flows can be defined much more generally. In this case the flows can be defined in the following way: Let γ be a closed convex N -sided polygon in the plane, and label the edges γ0, . . . , γN−1 in an anticlockwise order. Let θi ∈ S = R/2πZ be the angle of the exterior normal of γi, and let `i be the length of γi. Moving γ by a crystalline curvature flow consists of finding a continuous family of polygonal curves γ(t) starting from γ so that each edge keeps the same direction but moves in the outward normal direction with speed vi determined by its length: (1) vi(t) = gi(`i). Here gi is a smooth function defined on (0,∞) which is monotone increasing for each i. This paper mostly concerns contraction flows, for which gi 0 and fi is a positive real number for each i. 1991 Mathematics Subject Classification. 53C44, 52A10, 34C11. 1