arXivDaily arXiv每日学术速递 周一至周五更新
arXiv周末暂无论文更新，休息一下吧，周末愉快~~
arXiv 2201.10415math.DG
收藏

关于 S^4 中双调和 Clifford 环面的第二变分

On the second variation of the biharmonic Clifford torus in S^4

Stefano Montaldo, Cezar Oniciuc, Andrea Ratto

备注：
24 pages
发表信息：
Annals of Global Analysis and Geometry (2022) 62:791-814
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

AI总结：

本文研究 S^4 中真双调和 Clifford 环面的第二变分，计算其双调和指数与零化度，分析广义 Jacobi 算子核，并推广到极小浸入与双调和小超球面的复合。

摘要：

平坦环面 $\mathbb{T}=\mathbb{S}^1(\frac{1}{2})\times\mathbb{S}^1(\frac{1}{2})$ 可通过 $\Phi=i\circ\varphi$ 真双调和等距浸入到单位四维球面 $\mathbb{S}^4$ 中，其中 $\varphi:\mathbb{T}\to\mathbb{S}^3(\frac{1}{\sqrt2})$ 是极小 Clifford 环面，$i:\mathbb{S}^3(\frac{1}{\sqrt2})\to\mathbb{S}^4$ 是双调和小超球面。本文的首要目标是计算真双调和浸入 $\Phi$ 的双调和指数与零化度。随后，我们将详细研究广义 Jacobi 算子 $I_2^\Phi$ 的核。我们将证明该核包含一个方向，该方向容许一个自然变分，其一阶、二阶和三阶导数均为零，而四阶导数为负。在论文第二部分，我们将分析 $\varphi$ 对 $\Phi$ 的双调和指数与零化度的具体贡献。在此背景下，我们将研究更一般的复合 $\tilde\Phi=\tilde\varphi\circ i$，其中 $\tilde\varphi:M^m\to\mathbb{S}^{n-1}(\frac{1}{\sqrt2})$（$m\geq1$，$n\geq3$）是极小浸入，$i:\mathbb{S}^{n-1}(\frac{1}{\sqrt2})\to\mathbb{S}^n$ 是双调和小超球面。首先，我们将确定一个一般充分条件，以保证 $\tilde\Phi$ 的第二变分在 $\mathcal{C}(\tilde\varphi^{-1}T\mathbb{S}^{n-1})$ 上非负定。然后，我们在 Clifford 环面上完成此类分析，并作为补充结果得到 $\varphi$ 的 $p$-调和指数与零化度。在最后一节，我们将所得一般结果与可从等变第二变分研究中推出的结果进行比较。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑