Mean Curvature Flow and Nonlinear Heat Equations

平均曲率流是一个方程，它驱动表面在表面面积下降最陡的方向上的演变。当一个封闭的表面演变为尽可能快地缩小其面积时，凸点会向内移动，凹点会向外移动，并且表面越平坦，速度越慢。面积会减小，最终在有限的时间内趋于零。特别地，任何封闭曲面在有限时间内消失，因此奇点总是存在。这种流动起源于材料科学文献，并在纯数学和应用数学中得到了深入的研究。理解平均曲率流的关键是理解它所经过的奇点。这些项目关注于奇点的一些相关方面：(1)哪些奇点出现在一般流或一般流族中？(2)奇点附近的流动是什么样子的？什么时候爆炸是唯一的？(3)是否存在正则邻域定理？(4)奇异集的大小和结构是什么？什么时候奇异集是一个很好的子流形？(5)欧几里得三维空间中的奇点是否一般孤立？那么奇异时间呢？

Mean curvature flow is an equation that drives the evolution of a surface in the direction of steepest descent for the surface's area. As a closed surface evolves to decrease its area as rapidly as possible, convex points will move inwards, concave points move outwards, and the speed is slower where the surface is flatter. The area will decrease and eventually go to zero in finite time. In particular, any closed surface becomes extinct in finite time and, thus, singularities always occur. This flow originated in the materials science literature and has been intensely studied in both pure and applied mathematics. The key to understanding the mean curvature flow is to understand the singularities it goes through.These projects focus on a number of related aspects of the singularities: (1) Which singularities occur for a generic flow or a generic family of flows? (2) What does the flow look like near a singularity? When is the blow up unique? (3) Is there a canonical neighborhoods theorem? (4) What is the size and structure of the singular set? When is the singular set a nice submanifold? (5) Are singularities in Euclidean 3-space generically isolated? What about singular times?