平均曲率流的自相似解和奇性分析及其应用

Mean curvature flow is the negative gradient flow for volume, so any hypersurface flows through hypersurfaces in the direction of steepest descent for volume. Broadly speaking，the flow can be seen as a parabolic version of minimal submanifolds, hence it can be used to study problems in minimal submanifolds, even in low-dimensional topology and so on. The flow has been used and studied in material science for almost a century to model things like cell, grain, and bubble growth. .One of the most important problems in mean curvature flow is to understand the possible singularities that the flow goes through. It is very difficult to govern the structure of the flow near these singularities, and the resulting structure of the singular set. However, people have made significant progress for mean convex flow and flow with only generic singularities, and they also gave quite a few of interesting problems in this direction..As singularity models of the flow, self shrinkers act an important role in studying singularities of the flow. People define a geometric quantity: entropy for hypersurfaces in Euclidean space. A nature question is to classify all low entropy shrinkers, and a few of remarkable results have been archived in this direction. There are still some related problems, which are worth studying..Employing mean curvature flow, people have gotten sharp lower bounds on density for area-minimizing cones, and obtained classical solutions to the Dirichlet problem of minimal surface systems for a class of boundary maps. The flow can also be used to many other correlative problems.

平均曲率流是体积的负梯度流，它使超曲面沿体积下降最快的方向流动。它粗略的可看成极小子流形的抛物版本，可用来研究极小子流形，甚至低维拓扑等领域的问题。该流在材料科学中已使用、研究了近百年，用于模仿事物，如细胞、谷粒、气泡的增长。平均曲率流最重要的问题之一是研究流的奇点。掌握流在奇点附近的结构和它的奇点集的结构是很困难的。尽管如此，人们对平均凸和仅有‘一般性’奇点的流已有重要的认识，并提出了很多相关的值得研究的问题。作为奇点模型，自相似解在研究流的奇点中有重要的作用。人们对超曲面定义了几何量：熵，并在如何分类具有小熵的自收缩解上已取得了几个出色的结果，但仍有很多值得研究的地方。运用平均曲率流，人们证明了面积极小的锥的密度的最优下界，并给出了一类边界映照下极小曲面方程组的狄里克雷问题的经典解。该流还可以用来研究很多相关问题。

欧氏空间中子流形的平均曲率流是体积的负梯度流，它使子流形沿体积下降最快的方向流动。粗略的说，它是极小子流形的抛物版本，可用来研究极小子流形，甚至低维拓扑等领域的问题。该流在材料科学中已使用、研究了近百年，用于模仿事物，如细胞、谷粒、气泡的增长。平均曲率流最重要的问题之一是研究流的奇点。然而奇点的研究非常之困难，所以我们先研究奇点模型：自相似解，这对了解流的奇点有重要的意义。我们通过高斯映照、第二基本型的模长研究了自收缩解的刚性特征，得到了几个最优的刚性定理。通过平均曲率流自膨胀解研究了正则锥的扰动，特别是非面积极小的极小锥有正平均曲率光滑超曲面扰动；作为应用，可以研究非面积极小的极小锥的密度的无维数最优下界。通过研究从高余维图出发的平均曲率流长时间存在性，在一类边界条件下我们给出了极小曲面方程组的狄里克雷问题的经典解，这推广了Jenkins-Serrin经典结果；进一步我们还研究了解的边界正则性，其中我们的假设条件比前人弱很多。