  We study the Laplacian with zero magnetic field acting on complex functions
of a planar domain $\Omega$, with magnetic Neumann boundary conditions. If
$\Omega$ 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 $\Omega$. Besides the area, the perimeter
and the diameter, the geometric invariants which play a crucial role in the
estimates are the 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.
