论文标题
网络流中的0型奇异点 - 树木的演变
Type-0 singularities in the network flow -- Evolution of trees
论文作者
论文摘要
网络曲率运动的运动是对曲线缩短流的曲线有限的结合的概括。这种进化具有几个特征，主要是由于曲线相遇的结合存在。在本文中，我们表明，每当一条曲线的长度消失和两个三连接结合时，就会发展的网络的曲率保持界限。这种拓扑奇异性不包括网络流，可以称为0型奇异性，与众所周知的I型I和I型II相比，平滑曲线的通常平均曲率流或超曲面，其特征在于曲率的爆炸速率不同。结果，我们能够完整地描述类似树状网络的演变，直到第一个单数时间，假设所有切线流都具有单位多样性。如果这种溶液的寿命是有限的，则网络的曲率保持界限，我们可以通过Ilmanen-neves-Schulze/Lira-Mazzeo-Pluda-Saez应用结果，以在奇异之后重新启动流量。
The motion by curvature of networks is the generalization to finite union of curves of the curve shortening flow. This evolution has several peculiar features, mainly due to the presence of junctions where the curves meet. In this paper we show that whenever the length of one single curve vanishes and two triple junctions coalesce, then the curvature of the evolving networks remains bounded. This topological singularity is exclusive of the network flow and it can be referred as a Type-0 singularity, in contrast to the well known Type-I and Type-II ones of the usual mean curvature flow of smooth curves or hypersurfaces, characterized by the different rates of blow up of the curvature. As a consequence, we are able to give a complete description of the evolution of tree-like networks till the first singular time, under the assumption that all the tangents flows have unit multiplicity. If the lifespan of such solutions is finite, then the curvature of the network remains bounded and we can apply the results by Ilmanen-Neves-Schulze/Lira-Mazzeo-Pluda-Saez to restart the flow after the singularity.
