Evolving Plane Curves by Curvature in Relative Geometries
Evolving Plane Curves by Curvature in Relative Geometries
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DOI：
10.1215/s0012-7094-93-07216-x
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发表时间：
1993-11
影响因子：
2.5
通讯作者：
M. Gage;Yi A. Li
中科院分区：
文献类型：
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作者：
M. Gage;Yi A. Li
在 (0.1) 中，X :S × [0, ω) → IR 是一系列闭合凸平面曲线的位置向量，kN 是曲率向量，其中 k 是曲率，N 是由 N = -(cos θ, sin θ) 给出的向内指向法线。权重函数 γ(θ) = γ(N) 是曲线上每个点的法向向量的函数，但不依赖于平面中的位置。方程（0.1）有两个重要的解释。它可以被视为“曲线缩短”问题 ([Ga8]) 对 Minkowski 几何的推广，或者被视为金属晶体熔化时界面运动的简化模型 ([AnGu]、[Ta1] 和 [Ga8])。该证明说明了最近用于理解几何演化方程的大多数技术，如 [Ha3] 中所述。不难证明自相似解对应于方程的正 2π 周期解
In (0.1) X :S × [0, ω) → IR is the position vector of a family of closed convex plane curves, kN is the curvature vector, with k being the curvature and N the inward pointing normal given by N = −(cos θ, sin θ). The weight function γ(θ) = γ(N) is a function of the normal vector to the curve at each point but does not depend on position in the plane. Equation (0.1) has two significant interpretations. It can be seen as the generalization of the “curve shortening” problem ([Ga8]) to Minkowski geometry or as a simplified model of the motion of the interface of a metal crystal as it melts ([AnGu],[Ta1] and [Ga8]). The proof illustrates most of the techniques that have been used recently in understanding geometric evolution equations as described in [Ha3]. It is not hard to show that the self-similar solutions correspond to positive, 2π periodic solutions of the equation