Conformal invariants for the zero mode equation

For non-trivial solutions to the zero mode equation on a closed spin manifold $$D\phi =iA\cdot \phi ,$$ we first provide a simple proof for the sharp inequality \eq{ \norm{A}_{L^n}^2 \ge \frac {n}{4(n-1)} Y(M,[g]), } where $Y(M,[g])$ is the Yamabe constant of $(M,g)$, which was obtained by Frank-Loss and Reuss. Then we classify completely the equality case by proving that equality holds if and only if $\phi$ is a Killing spinor, and if and only if $(M,g)$ is a Sasaki-Einstein manifold with $A$ (up to scaling) as its Reeb field and $\phi$ a vacuum up to a conformal transformation. More generalizations have been also studied.