标题: | n维单位球面上威尔摩曲面保角类的夹挤定理 Pinching Theorems for Conformal Classes of Willmore Surfaces in the Unit N-sphere |
作者: | 张有中 Yu-Chung Chang 许义容 Yi-Jung Hsu 应用数学系所 |
关键字: | 威尔摩曲面;球面;保角类;Willmore surface;sphere;conformal class |
公开日期: | 2004 |
摘要: | 令x是一个从紧致威尔摩曲面M到n 维单位球面Sn的浸入,在本论文中我们首先考虑3维单位球面中的威尔摩曲面并建立一个包含第二基本型式迹为零部份的张量长度平方与平均曲率之积分不等式。基于此积分不等式,我们可藉着某些夹挤的条件对全脐球面与Clifford环面进行分类。接着我们介绍一个由第二基本型式迹为零部份的张量长度平方与平均曲率形成的保角不变量,并证明此保角不变量的上界若为Clifford环面之值,则x(M)不是全脐球面就是Clifford环面的保角类。我们同样考虑n维单位球面中的威尔摩曲面,并藉着某些夹挤的条件对全脐球面与Veronese曲面进行分类。最后我们模仿3维的情况,引进一个保角不变量,并证明此保角不变量的上界若为Veronese曲面之值,则x(M)不是全脐球面就是Veronese曲面的保角类。 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. |
URI: | http://140.113.39.130/cdrfb3/record/nctu/#GT008822802 http://hdl.handle.net/11536/63557 |
显示于类别: | Thesis |
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