多相均曲率流的自相似擴張解

Abstract

我們考慮由有限多個通過原點的曲面所組成的多相曲面C0,其中所有的一維結線是由三曲面兩兩夾角相同所組合形成的。此外各曲面對原點做徑向投影所形成的曲線也是有限長。我們證明了以此C0做初始條件下存在一多相均曲率流的自相似擴張解,而這組解是由那些正規三結線和正規四節點所組合出來的曲面。

We consider a multiphase surface C0 in R3 consisting of a finite number of surfaces passing through the origin , where all 1-dimensional junctions are regular triple junctions in which three planes meet at the same angle and each surface scales down homothetically to a limit curve of finite length. We prove the existence of self-similar expanding solutions of the mean curvature flow on the multiphase surface initially given by C0. For this initial C0, there are multiple solutions that are combinations of the regular triple junctions and regular quadruple points, where four regular triple junctions meet at an angle of approximately 109.5◦.