Spinor inequality for magnetic fields on spin manifolds

This paper is concerned with the zero mode equation $D_g\phi =iA\cdot \phi$ on closed spin manifold $(M^n,g,\sigma )$ of positive scalar curvature. Here $A$ is a real one form on $M$. We proved that if $(\phi ,A)$ is a non trivial solution of the zero mode equation then $$\parallel dA{\parallel }_{n\mathrm{/}2}>Y(M^n,[g])\mathrm{/}(4v_n^{1\mathrm{/}2}),$$ where $Y(M^n,[g])$ is the Yamabe constant of $(M^n,g)$ and $v_n=\left[\frac{n}{2}\right]$. In the case of the round sphere $({\mathbb{S}}^n,g_{can},{\sigma }_{can})$ this result confirms that the inequality obtained in \cite{Frank} is not sharp.