逆平均曲率流之研究

Abstract

這篇文章中，我們討論超曲面在對曲率做合理假設下的旋轉對稱空間中沿著逆平均曲率流之行為。我們針對逆平均曲率流的初始曲面為封閉、星形且mean-convex之情形的長時間存在性以及近似行為做細部分析。另外我們利用逆平均曲率流來證明歐氏空間中定義域為星形且mean-convex之quermassintegrals的isoperimetric不等式。

In this thesis, we study the behavior of the motion of hypersurfaces by their inverse mean curvature flow (abbreviated as IMCF) in the rotational symmetric space with reasonable condition on its curvatures. In particular, we give a detailed analysis about the long time existence and the asymptotic behavior of the IMCF when the initial surface is closed star-shaped and mean-convex. We also present an application of the IMCF to the proof of the isoperimetric inequality for quermassintegrals of mean-convex star-shaped domains in Euclidean space.