平面网络流的自相似展开解
Self similar expanding solutions of the planar network flow
摘要：
本文证明了平面网络上曲率流的自相似展开解的存在性，其中初始构形是任意多个在原点相交的半直线。这概括了施努尔和舒尔茨最近的工作，其中对待三个半线的情况。有多个解决方案，这些是参数化的组合对象，即斯坦纳树相对于一个完整的负弯曲度量的单位球上的跨度$k$指定点的边界上的无穷大。我们还提供了一个尖锐的制定这些解决方案的规律性在$t=0$。
英文摘要：
We prove the existence of self-similar expanding solutions of the curvature flow on planar networks where the initial configuration is any number of half-lines meeting at the origin. This generalizes recent work by Schnürer and Schulze which treats the case of three half-lines. There are multiple solutions, and these are parametrized by combinatorial objects, namely Steiner trees with respect to a complete negatively curved metric on the unit ball which span $k$ specified points on the boundary at infinity. We also provide a sharp formulation of the regularity of these solutions at $t=0$.
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