On Willmore Legendrian surfaces in S 5 S5 and the contact stationary Legendrian Willmore surfaces
On Willmore Legendrian surfaces in S 5 S5 and the contact stationary Legendrian Willmore surfaces
复制标题
在 S 5 S5 中的 Willmore Legendrian 曲面和接触固定 Legendrian Willmore 曲面上
DOI：
10.1007/s00526-017-1183-z
复制
发表时间：
2016
影响因子：
2.1
通讯作者：
Yong Luo
中科院分区：
文献类型：
--
作者：
Yong Luo
在本文中，我们研究Willmore Legendrian曲面（即Legendrian曲面，这是临界点的Willmore功能）。我们使用一个在Luo中证明的等式（ arXiv：1211.4227v6 ）得到了中Willmore Legendrian曲面与接触平稳Legendrian曲面之间的一个关系，并利用这个关系证明了中Willmore Legendrian球面的一个分类结果.我们还得到了Willmore Legendrian曲面的一个积分不等式，特别证明了如果Willmore Legendrian曲面的第二基本形式的平方长度属于[0，2]，则它必为0且L是全测地线或2且L是平坦的极小Legendrian环面，推广了Yamaguchi等人（Proc Am Math Soc 54：276-280，1976）的结果.我们还研究了5维Sasakian流形中Legendrian曲面之间Willmore泛函的变化。设是一个闭曲面和一个具有切触形式、度量和几乎复结构的5维Sasakian流形J.假设这是一个勒让德式浸入。如果接触形变下的Willmore泛函的临界点是接触平稳LegendrianWillmore曲面（简称csLWillmore曲面），则称之为接触平稳LegendrianWillmore曲面。为了研究csL Willmore曲面的存在性，我们引入了一个高阶流，它保持了Legendre条件并降低了Willmore能量。作为第一步，我们证明了这个流是适定的ifis一个Sasakian爱因斯坦流形，特别是。
In this paper we study Willmore Legendrian surfaces (that is Legendrian surfaces which are critical points of the Willmore functional). We use an equality proved in Luo ( arXiv:1211.4227v6 ) to get a relation between Willmore Legendrian surfaces and contact stationary Legendrian surfaces in, and then we use this relation to prove a classification result for Willmore Legendrian spheres in. We also get an integral inequality for Willmore Legendrian surfaces and in particular we prove that if the square length of the second fundamental form of a Willmore Legendrian surface inbelongs to [0, 2], then it must be 0 andLis totally geodesic or 2 andLis a flat minimal Legendrian tori, which generalizes the result of Yamaguchi et al. (Proc Am Math Soc 54:276–280, 1976). We also study variation of the Willmore functional among Legendrian surfaces in 5-dimensional Sasakian manifolds. Letbe a closed surface anda 5-dimensional Sasakian manifold with a contact form, an associated metricand an almost complex structureJ. Assume thatis a Legendrian immersion. Thenfis called a contact stationary Legendrian Willmore surface (in short, a csL Willmore surface) if it is a critical point of the Willmore functional under contact deformations. To investigate the existence of csL Willmore surfaces we introduce a higher order flow which preserves the Legendre condition and decreases the Willmore energy. As a first step we prove that this flow is well posed ifis a Sasakian Einstein manifold, in particular.