Lower bounds for the first eigenvalue of the magnetic Laplacian

We consider a Riemannian cylinder $\Omega$ endowed with a closed potential $1$-form $A$ and study the magnetic Laplacian $\Delta_A$ with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate.