二维自扩张子的唯一性  

The uniqueness of 2-dimensional self-expanders

作　　者：富宇[1] 莫小欢[2] Yu Fu;Xiaohuan Mo

机构地区：[1]东北财经大学数据科学与人工智能学院,大连116025 [2]北京大学数学科学学院,北京100871

出　　处：《中国科学：数学》2024年第10期1533-1544,共12页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:12171005);辽宁省兴辽英才计划(批准号:XLYC2203194);辽宁省教育厅项目(批准号:LJKMZ20221561)资助项目。

摘　　要：平均曲率流中最重要的问题之一是研究其可能出现的奇点.平均曲率流的自相似解不仅与曲率流的奇点密切相关,而且也是一类重要的子流形.本文研究平均曲率流自相似解的分类问题,特别是通过发展求解偏微分方程的新技巧来研究3维Euclid空间中自扩张子的唯一性问题.本文证明了对于3维Euclid空间R^(3)中的二维自扩张子X:M→R^(3),如果其Gauss曲率K是常数,则K一定为0,而且自扩张子局部只能是柱面Γ×R,其中Γ为R^(2)中的自扩张子;进一步证明了如果二维自扩张子X的第二基本形式模长平方是常数,则自扩张子只能是平面R^(2).One of the most important problems in the study of the mean curvature fows is to investigate their possible singularities.The self-similar solutions of the mean curvature fows are not only closely related to the singularities of the fows but also one important class of submanifolds.In this paper,the classification of selfsimilar solutions for the mean curvature fows is studied.In particular,by developing new techniques for solving partial differential equations,we study the uniqueness of self-expanders in the 3-dimensional Euclidean space.It is proved that for the 2-dimensional self-expander X:M→R^(3) in the 3-dimensional Euclidean space R^(3),if the Gauss curvature K is constant,then K must be vanishing,and locally X:M→R^(3) is Γ×R,where Γ is the self-expander curve in R^(2).Furthermore,it is shown that X:M→R^(3) must be a plane R^(2) provided that the squared norm of the second fundamental form is constant.

关 键 词：平均曲率流 自相似解 自扩张子 

分 类 号：O186.1[理学—数学]