歐氏空間裡超曲面的第二基本型的泛函的第一變分公式

Abstract

在歐氏空間 R^(n+1) 裡,超曲面 M^n 的體積的變分公式是我們所熟知的. 在這篇論文一開始 我們將導出在歐氏空間裡,超曲面第二基本型的泛函的第一變分公式. 我們知道廣義的 Willmore 泛函在 保角變換底下是不變的. 在第三節我們試著找出其他有趣的,在保角變換底下不變的泛函. 在最後一節,我們利用第一節找出廣義的 Willmore 泛函的第一變分公式.

The first variational formula of the volume for a hypersurface M^n in the Euclidean space R^(n+1) is well-known. In this Master's dissertation we begin by deriving the first variational formula of the functional of the traces of the second fundamental form for hypersurfaces in the Euclidean space. We know that a generalized Willmore functional is invariant under conformal mapping. In section 3, we try to find another interesting functionals which are invariant under conformal transformations. In final section, using section 1, we find the first variational formula of a generalized Willmore functional.