Another sharp criterion for Dirac zero modes

Abstract: We resolve an open problem posed by Frank--Loss. Let $n\ge 3$ and let $\varphi\in L<sup>p(\mathbb{S}<sup>n)$, with $\frac{n}{n-1}&lt;p&lt;\infty$, be a nontrivial Dirac zero mode on $\mathbb{S}^n$, i.e. a nonzero spinor satisfying \begin{equation*} D\varphi = iA\cdot\varphi, \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, $\varphi$ is a Killing spinor and $A$ is a real multiple of the Reeb field associated with $\varphi$. The case $n=3$ was proved in our 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.