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. (C) 2018 Elsevier Inc. All rights reserved.