Lower bounds for the first eigenvalue of the Laplacian with zero magnetic field in planar domains

We study the Laplacian with zero magnetic field acting on complex functions of a planar domain Ω, with magnetic Neumann boundary conditions. If Ω is simply connected then the spectrum reduces to the spectrum of the usual Neumann Laplacian; therefore we focus on multiply connected domains bounded by convex curves and prove lower bounds for its ground state depending on the geometry and the topology of Ω. Besides the area, the perimeter and the diameter, the geometric invariants which play a crucial role in the estimates are the fluxes of the potential one-form around the inner holes and the distance between the boundary components of the domain; more precisely, the ratio between its minimal and maximal width. Then, we give a lower bound for doubly connected domains which is sharp in terms of this ratio, and a general lower bound for domains with an arbitrary number of holes. When the inner holes shrink to points, we obtain as a corollary a lower bound for the first eigenvalue of the so-called Aharonov-Bohm operators with an arbitrary number of poles.