论文标题
在S^4中的Biharmonic Clifford圆环的第二个变化中
On the second variation of the biharmonic Clifford torus in S^4
论文作者
论文摘要
扁平圆环$ {\ Mathbb t} = {\ Mathbb s}^1 \ left（\ frac {1} {1} {2} {2} \ right）\ times {\ times {\ mathbb s}^1 \ left（\ frac {1} {1} {1} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} {2} $ {\ mathbb s}^4 $由$φ= i \circcφ$给出，其中$φ：{\ Mathbb t} \ to {\ Mathbb s}^3（\ frac {1} {\ sqrt 2}） s}^3（\ frac {1} {\ sqrt 2}）\ to {\ mathbb s}^4 $是Biharmonic Small hypersphere。本文的第一个目标是计算适当的Biharmonic Immersion $φ$的Biharmonic指数和无效。之后，我们将详细研究广义雅各比操作员$ i_2^φ$的内核。我们将证明它包含一个方向，该方向在第一，第二和第三个衍生物消失的方向上接受了自然变化，因此第四个导数为负。在本文的第二部分中，我们将分析$φ$对Biharmonic指数的特定贡献和$φ$的无效。在这种情况下，我们将研究更一般的组成$ \tildeφ= \tildeφ\ circ i $，其中$ \tildeφ：m^m \ to {\ mathbb s}^{n-1} {n-1}（\ frac {1} {1}和$ i：{\ mathbb s}^{n-1}（\ frac {1} {\ sqrt 2}）\ to {\ mathbb s}^n $是Biharmonic的小型大型球。首先，我们将确定一种一般的足够条件，该条件确保$ \tildeφ$的第二个变化在$ \ m varycal {c} \ big（\tildeφ^{ - 1} t {\ mathbb s}^{n-1}^{n-1} \ big）上。然后，我们完成对Clifford torus的这种分析，作为互补的结果，我们获得了$ p $ harmonic索引和$φ$的无效。在最后一部分中，我们将一般结果与可以从模棱两可的第二个变化的研究中得出的结果进行了比较。
The flat torus ${\mathbb T}={\mathbb S}^1\left (\frac{1}{2} \right ) \times {\mathbb S}^1\left (\frac{1}{2} \right )$ admits a proper biharmonic isometric immersion into the unit $4$-dimensional sphere ${\mathbb S}^4$ given by $Φ=i \circ φ$, where $φ:{\mathbb T} \to {\mathbb S}^3(\frac{1}{\sqrt 2})$ is the minimal Clifford torus and $i:{\mathbb S}^3(\frac{1}{\sqrt 2}) \to {\mathbb S}^4$ is the biharmonic small hypersphere. The first goal of this paper is to compute the biharmonic index and nullity of the proper biharmonic immersion $Φ$. After, we shall study in the detail the kernel of the generalised Jacobi operator $I_2^Φ$. We shall prove that it contains a direction which admits a natural variation with vanishing first, second and third derivatives, and such that the fourth derivative is negative. In the second part of the paper we shall analyse the specific contribution of $φ$ to the biharmonic index and nullity of $Φ$. In this context, we shall study a more general composition $\tildeΦ=\tildeφ \circ i$, where $\tildeφ: M^m \to {\mathbb S}^{n-1}(\frac{1}{\sqrt 2})$, $ m \geq 1$, $n \geq {3}$, is a minimal immersion and $i:{\mathbb S}^{n-1}(\frac{1}{\sqrt 2}) \to {\mathbb S}^n$ is the biharmonic small hypersphere. First, we shall determine a general sufficient condition which ensures that the second variation of $\tildeΦ$ is nonnegatively defined on $\mathcal{C}\big (\tildeφ^{-1}T{\mathbb S}^{n-1}\big )$. Then we complete this type of analysis on our Clifford torus and, as a complementary result, we obtain the $p$-harmonic index and nullity of $φ$. In the final section we compare our general results with those which can be deduced from the study of the equivariant second variation.
