论文标题
数值计算平均曲率流量自我脱离器的索引
Numerically computing the index of mean curvature flow self-shrinkers
论文作者
论文摘要
通过平均曲率流发展的表面会发展出奇异性。这些奇异性可以通过自我碎裂的人来建模，这些表面是通过流量下的扩张而收缩的。以经典的自我撕裂器（即球形和圆柱体）为模型的奇异性在流动的扰动下是稳定的。相比之下，以其他自我缩减器（例如天形圆环）为模型的奇异性是不稳定的：扰动流动通常会改变奇异性的种类。人们可以通过计算自我缩减器的摩尔斯索引来衡量不稳定性的程度，这被视为适当功能的关键点。 在本文中，我们提出了一种用于计算旋转对称自我撕裂者索引的数值方法。我们将此方法应用于绿色的圆环，这是自我撕裂者的第一个已知的非平凡示例。我们发现，除了扩张和翻译外，卷轴的索引为$ 5 $，这与我们早期工作的Liu工作中的$ 3 $的下限和29美元的上限一致。另外，我们出乎意料地发现了属于欧文纳$ -1 $的绿色圆环的另外两种变体。
Surfaces that evolve by mean curvature flow develop singularities. These singularities can be modeled by self-shrinkers, surfaces that shrink by dilations under the flow. Singularities modeled on classical self-shrinkers, namely spheres and cylinders, are stable under perturbations of the flow. In contrast, singularities modeled on other self-shrinkers, such as the Angenent torus, are unstable: perturbing the flow will generally change the kind of singularity. One can measure the degree of instability by computing the Morse index of the self-shrinker, viewed as a critical point of an appropriate functional. In this paper, we present a numerical method for computing the index of rotationally symmetric self-shrinkers. We apply this method to the Angenent torus, the first known nontrivial example of a self-shrinker. We find that, excluding dilations and translations, the index of the Angenent torus is $5$, which is consistent with the lower bound of $3$ from the work of Liu and the upper bound of $29$ from our earlier work. Also, we unexpectedly discover two additional variations of the Angenent torus with eigenvalue $-1$.
