发表机构
Albert-Ludwigs-Universität Freiburg; Wuhan University（弗莱堡大学; 武汉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究解决了Frank-Loss提出的关于球面上Dirac零模的尖锐下界问题，证明了一个普适不等式，并指出等号仅在奇数维时可达，此时模去共形与规范变换后为Killing旋量；对n≥5的情形，论证归结为重心约束下的改进Sobolev不等式，并给出其普适估计。
AI 中文摘要
我们解决了Frank--Loss提出的一个开放问题。设$n\ge 3$，且$\varphi\in L^p(\mathbb{S}^n)$，其中$\frac{n}{n-1}<p<\infty$，是$\mathbb{S}^n$上的非平凡Dirac零模，即满足\begin{equation*} D\varphi = iA\cdot\varphi, \end{equation*}的非零旋量，其中$D$是Dirac算子，$A$是满足$\mathrm{d} A^\flat\in L^{n/2}$的向量场。我们证明尖锐下界\begin{equation*} \\|\mathrm{d} A^\flat\\|_{\frac{n}{2}} \ge 2\left[\frac{n}{2}\right]^{-\frac12}\frac{n-1}{n-2}S_n = \left[\frac{n}{2}\right]^{-\frac12}\frac{n(n-1)}{2}\omega_n^{\frac{2}{n}}. \end{equation*}等号可达当且仅当$n$为奇数；在这种情况下，模去共形和规范变换，$\varphi$是Killing旋量，$A$是与$\varphi$关联的Reeb场的实倍数。$n=3$的情形在我们最近的论文中用不同方法证明。在本文中，我们将剩余情形分为3种：i) $n\ge 5$为奇数，ii) $n\ge 5$为偶数，iii) $n=4$。所有这些情形都需要使用不同的方法。对于$n\ge 5$，论证关键归结为在重心约束下$\mathbb{S}^n$上的改进Sobolev不等式。具体地，对于$u\in W^{1,2}(\mathbb{S}^n)$，我们考虑\begin{equation*} \mathfrak{a}_n:= \inf\Bigg\{ \frac{ \int \Big(|\nabla u|^2 + \frac{n(n-2)}{4}u^2\Big) - \frac{n(n-2)}{4}\omega_n^{\frac{2}{n}}\\|u\\|_{\frac{2n}{n-2}}^2 }{ \omega_n^{\frac{2}{n}}\\|u\\|_{\frac{2n}{n-2}}^2 - \int u^2 } \\,\Bigg|\\, \int x|u|^{\frac{2n}{n-2}}=0,\\ \omega_n^{\frac{2}{n}}\\|u\\|_{\frac{2n}{n-2}}^2 - \int u^2>0 \Bigg\}, \end{equation*}并得到以下普适估计\begin{equation*} \mathfrak{a}_n > \frac{n(n-2)}{4(n^2-3n+1)}, \end{equation*}这足以达到我们的目的。确定$\mathfrak{a}_n$的精确值仍然开放。
英文摘要
We resolve an open problem posed by Frank--Loss. Let $n\ge 3$ and let $φ\in L^p(\mathbb{S}^n)$, with $\frac{n}{n-1}<p<\infty$, be a nontrivial Dirac zero mode on $\mathbb{S}^n$, i.e. a nonzero spinor satisfying \begin{equation*} Dφ= iA\cdotφ, \end{equation*} where $D$ is the Dirac operator and $A$ is a vector field with $\mathrm{d} A^\flat\in L^{n/2}$. We prove the sharp lower bound \begin{equation*} \|\mathrm{d} A^\flat\|_{\frac{n}{2}} \ge 2\left[\frac{n}{2}\right]^{-\frac12}\frac{n-1}{n-2}S_n = \left[\frac{n}{2}\right]^{-\frac12}\frac{n(n-1)}{2}ω_n^{\frac{2}{n}}. \end{equation*} Equality is attainable if and only if $n$ is odd; in that case, modulo conformal and gauge transformations, $φ$ is a Killing spinor and $A$ is a real multiple of the Reeb field associated with $φ$. The case $n=3$ was proved in our recent paper using a different method. In the paper we divide the remaining cases into 3 cases: i) $n\ge 5$ is odd, ii) $n\ge 5$ is even and iii) $n=4$. All these cases need to use different methods. For $n\ge 5$, the argument crucially reduces to an improved Sobolev inequality on $\mathbb{S}^n$ under a barycenter constraint. Specifically, for $u\in W^{1,2}(\mathbb{S}^n)$ we consider \begin{equation*} \mathfrak{a}_n := \inf\Bigg\{ \frac{ \int \Big(|\nabla u|^2 + \frac{n(n-2)}{4}u^2\Big) - \frac{n(n-2)}{4}ω_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 }{ ω_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 - \int u^2 } \,\Bigg|\, \int x|u|^{\frac{2n}{n-2}}=0,\ ω_n^{\frac{2}{n}}\|u\|_{\frac{2n}{n-2}}^2 - \int u^2>0 \Bigg\}, \end{equation*} and we obtain the following universal estimate \begin{equation*} \mathfrak{a}_n > \frac{n(n-2)}{4(n^2-3n+1)}, \end{equation*} which is enough for our aim. Determining the exact value of $\mathfrak{a}_n$ remains open.
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