We consider a Riemannian cylinder endowed with a closed potential 1-form A
and study the magnetic Laplacian 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.Comment: Replaces in part arXiv:1611.0193
