论文标题
旋转对称的自我缩减摩托车索引的界限
Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori
论文作者
论文摘要
在有限的时间内，在平均曲率流下进化的封闭表面变为单数。在奇异性附近，表面类似于一个自我碎裂器，这种表面在平均曲率流下通过扩张而缩小。如果奇异性是在除圆形或圆柱体以外的自我缩合器上建模的，则在流动的扰动下，奇异性是不稳定的。当被视为熵功能的临界点时，可以使用自我撕裂器的索引来量化这种不稳定性。 在这项工作中，我们证明了旋转对称的自碎托里索引的索引以及最大和最小半径的上限。尽管文献中已经有一些下限结果，但我们认为，这是自我缩减器索引上的第一个上限。我们的方法还给出了索引和熵的下限，我们的方法为两个熵折扣变化提供了简单的公式，其存在由Liu证明。令人惊讶的是，与这些变化相对应的特征值正好为$ -1 $。最后，我们在更高的维度和六个潜在方向上提供了一些初步结果。
A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using the index of the self-shrinker when viewed as a critical point of the entropy functional. In this work, we prove an upper bound on the index of rotationally symmetric self-shrinking tori in terms of their entropy and their maximum and minimum radii. While there have been a few lower bound results in the literature, we believe that this result is the first upper bound on the index of a self-shrinker. Our methods also give lower bounds on the index and the entropy, and our methods give simple formulas for two entropy-decreasing variations whose existence was proved by Liu. Surprisingly, the eigenvalue corresponding to these variations is exactly $-1$. Finally, we present some preliminary results in higher dimensions and six potential directions for future work.
