A Characterization of the Ejiri Torus in S-5
A Characterization of the Ejiri Torus in S-5
复制标题
S-5 中 Ejiri 环面的表征
DOI：
10.1007/s10114-016-5491-6
复制
发表时间：
2016
影响因子：
0.7
通讯作者：
Wang Peng
中科院分区：
文献类型：
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作者：
Wang Peng
我们猜想，Willmore泛函在2π 2和2π 2之间的Willmore环面要么共形等价于Clifford环面，要么共形等价于Ejiri环面. S5中的Ejiri环面是Willmore曲面的第一个例子，它不与任何真实的空间形式中的极小曲面共形等价。Li和Vrancken通过将Sn中张量积的Willmore曲面约化为S3中的弹性曲线，对Sn中的所有Willmore曲面进行了分类，Ejiri环面作为特例出现。本文首先证明了在所有张量积Willmore曲面中，S5中Ejiri曲面的Willmore泛函达到最小值2π2，这表明我们的猜想对张量积Willmore曲面成立.证明了当余维数足够大时，张量积的Willmore环面都是不稳定的。我们还表明，Ejiri环面是不稳定的，即使在S5。此外，类似于Li和Vrancken，我们通过在S3中用弹性曲线约化张量积的所有约束Willmore曲面来分类它们。当余维足够大时，所有这样得到的约束Willmore环面也是不稳定的。
We conjecture that a Willmore torus having Willmore functional between 2π2and 2π2is either conformally equivalent to the Clifford torus, or conformally equivalent to the Ejiri torus. Ejiri’s torus inS5is the first example of Willmore surface which is not conformally equivalent to any minimal surface in any real space form. Li and Vrancken classified all Willmore surfaces of tensor product inSnby reducing them into elastic curves inS3, and the Ejiri torus appeared as a special example. In this paper, we first prove that among all Willmore tori of tensor product, the Willmore functional of the Ejiri torus inS5attains the minimum 2π2, which indicates our conjecture holds true for Willmore surfaces of tensor product. Then we show that all Willmore tori of tensor product are unstable when the co-dimension is big enough. We also show that the Ejiri torus is unstable even inS5. Moreover, similar to Li and Vrancken, we classify all constrained Willmore surfaces of tensor product by reducing them with elastic curves inS3. All constrained Willmore tori obtained this way are also shown to be unstable when the co-dimension is big enough.