Brakke流的爆破方法：三结点附近的网络
The blow up method for Brakke flows: networks near triple junctions
摘要：
本文引入一种抛物Blow-up方法来研究平面网络的积分Brakke流（即二维区域中的一维积分Brakke流）在时空区域中弱接近静态重数为1的三重结点$J$的渐近行为.我们表明，这样的网络流是定期在一个较小的时空区域，在这个意义上说，它由三条曲线来顺利地在一个点在120度角，保持顺利接近$J$和顺利移动。利用这个结果和White的分层定理，我们推出：当时空区域${\mathcal R}$中的网络的积分Brakke流不存在密度为$\geq2$的静态切流时，存在一个抛物豪斯多夫维数至多为1的闭子集$Σ\subset {\mathcal R}$，使得该流在${\mathcal R} \setminus Σ$中是经典流，即在${\mathcal R} \setminus Σ$中的每一点附近，如果该流非空，由平滑移动的嵌入曲线或在120度角的单个点处平滑相交并平滑移动的三条嵌入曲线组成。特别地，这样的流在任何时候都是经典的，除了最多$\frac{1}{2}$的普通豪斯多夫维数的闭时间集。
英文摘要：
We introduce a parabolic blow-up method to study the asymptotic behavior of an integral Brakke flow of planar networks (i.e. a 1-dimensional integral Brakke flow in a two dimensional region) weakly close in a space-time region to a static multiplicity 1 triple junction $J$. We show that such a network flow is regular in a smaller space-time region, in the sense that it consists of three curves coming smoothly together at a single point at 120 degree angles, staying smoothly close to $J$ and moving smoothly. Using this result and White's stratification theorem, we deduce that whenever an integral Brakke flow of networks in a space-time region ${\mathcal R}$ has no static tangent flow with density $\geq2$, there exists a closed subset $Σ\subset {\mathcal R}$ of parabolic Hausdorff dimension at most 1 such that the flow is classical in ${\mathcal R} \setminus Σ$, i.e. near every point in ${\mathcal R} \setminus Σ$, the flow, if non-empty, consists of either an embedded curve moving smoothly or three embedded curves meeting smoothly at a single point at 120 degree angles and moving smoothly. In particular, such a flow is classical at all times except for a closed set of times of ordinary Hausdorff dimension at most $\frac{1}{2}$.
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