基於羅森－歐斯曼錐構造的均曲流自相似解

Self-similar solutions to the mean curvature flow based on the Lawson-Osserman cone

李宸寬(Chen-Kuan Lee)

指導教授 ： 蔡忠潤

國立臺灣大學/理學院/數學研究所/碩士(2022年)

https://doi.org/10.6342/NTU202200888

摘要

在這篇論文中，我們首先得到基於羅森－歐斯曼錐構造的均曲流自相似解必須滿足的等式，並證明了自擴張解的存在性。主要的關鍵是利用羅森－歐斯曼錐的對稱性將偏微分方程轉化為常微分方程組，並研究這種近似於自治系統的常微分方程組。特別地，我們發現從狄利克雷問題的觀點來看，我們構造的自擴張解具唯一性。

關鍵字

幾何分析 ； 高餘維均曲流 ； 自相似解 ； 羅森－歐斯曼錐 ； 狄利克雷問題

並列摘要

In this thesis, we derived the equation of self-similar solutions to mean curvature flow based on the Lawson-Osserman cone and proved the existence of self-expander. The main point is to use the symmetry to transform the PDE into a system of ODEs and analyze such analogous autonomous system. In particular, the self-expander is unique form the viewpoint of Dirichlet problem.

並列關鍵字

Geometric Analysis ； Mean Curvature Flow in Higher Codimensions ； Self-Similar Solution ； Lawson-Osserman Cone ； Dirichlet Problem

參考文獻

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