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We obtain upper bounds for the first eigenvalue of the magnetic Laplacianassociated to a closed potential $1$-form (hence, with zero magnetic field)acting on complex functions of a planar domain $\Omega$, with magnetic Neumannboundary conditions. It is well-known that the first eigenvalue is positivewhenever the potential admits at least one non-integral flux. By gaugeinvariance the lowest eigenvalue is simply zero if the domain is simplyconnected; then, we obtain an upper bound of the ground state energy dependingonly on the ratio between the number of holes and the area; modulo a numericalconstant the upper bound is sharp and we show that in fact equality is attained(modulo a constant) for Aharonov-Bohm-type operators acting on domainspunctured at a maximal $\epsilon$-net. In the last part we show that the upperbound can be refined, provided that one can transform the given domain in asimply connected one by performing a number of cuts with sufficiently smalltotal length; we thus obtain an upper bound of the lowest eigenvalue by theratio between the number of holes and the area, multiplied by a Cheeger-typeconstant, which tends to zero when the domain is metrically close to a simplyconnected one.

Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains