n維單位球面上威爾摩曲面保角類的夾擠定理

Abstract

令x是一個從緊緻威爾摩曲面M到n 維單位球面Sn的浸入，在本論文中我們首先考慮3維單位球面中的威爾摩曲面並建立一個包含第二基本型式跡為零部份的張量長度平方與平均曲率之積分不等式。基於此積分不等式，我們可藉著某些夾擠的條件對全臍球面與Clifford環面進行分類。接著我們介紹一個由第二基本型式跡為零部份的張量長度平方與平均曲率形成的保角不變量，並證明此保角不變量的上界若為Clifford環面之值，則x(M)不是全臍球面就是Clifford環面的保角類。我們同樣考慮n維單位球面中的威爾摩曲面，並藉著某些夾擠的條件對全臍球面與Veronese曲面進行分類。最後我們模仿3維的情況，引進一個保角不變量，並證明此保角不變量的上界若為Veronese曲面之值，則x(M)不是全臍球面就是Veronese曲面的保角類。

Let x be an immersion of a compact Willmore surface M into the n-dimensional unit sphere Sn. In this thesis we first consider the Willmore surfaces in the unit 3-sphere, and establish an integral inequality for the square of the length of the trace free part of the second fundamental form and the mean curvature. Based on this integral inequality, we characterize the totally umbilical spheres and the Clifford torus by a certain pinching condition. We then introduce a conformal invariant quantity which is formulated in terms of the square of the length of the trace free part of the second fundamental form and the mean curvature, and prove that if this quantity is bounded above by that value of the Clifford torus then x(M) is either a totally umbilical sphere or a conformal Clifford torus. As for the case n = 3; we also characterize the totally umbilical spheres and the Veronese surface by a pinching condition for the case n≧4: Analogous to the case n = 3; we then introduce a conformal invariant quantity, and prove that if this quantity is bounded above by that value of the Veronese surface then x(M) is either a totally umbilical sphere or a conformal Veronese surface.