高餘維子流形的存在性和唯一性

Abstract

高餘維極小子流形有很多性質是不同於超曲面的情況。文章中我們主要探討高餘維極小子流形的存在性和唯一性問題。 論文前半部將探討圖形式 (graphical) 極小子流形的唯一性問題。圖形式極小子流形滿足的微分方程稱之為極小曲面方程 (minimal surface system). 根據 [LO77], 極小曲面方程的狄利克雷問題 (Dirichlet problem) 的解一般並不會有唯一性。我們通過凸優化理論的工具，證明了當狄利克雷問題的解的奇異值 (singular value) 滿足特定條件時，它就會有唯一性。 論文後半部討論有奇異點的極小子流形的存在性問題。我們推廣了 [CHS84] 關於帶奇異點極小超曲面的結果到高餘維度的情況。我們證明了給定正則的極小錐，可以在它附近找到帶奇異點非錐的高餘維極小子流形。

Higher codimension minimal submanifold behaves differently from the hypersurface case. In this thesis, we focus on the aspect of existence and uniqueness of higher codimension minimal submanifold. The first part of this thesis discusses the uniqueness property of graphical minimal submanifold. The uniqueness of the Dirichlet problem of minimal surface system is generally false as noted by [LO77]. By using tools from convex optimization theory, we are able to give sufficient condition in term of singular value for the uniqueness problem. The second part discusses the existence of general minimal submanifold with isolated singularity in higher codimension but is not a cone. We generalize the Leray fix-point method in [CHS84] to higher codimension setting and show the existence of non-conical higher codimension minimal submanifold (with boundary) with isolated singularity.