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

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