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We discuss isoperimetric inequalities for the magnetic Laplacian on boundeddomains of $\mathbb R^2$ endowed with an Aharonov-Bohm potential. When the fluxof the potential around the pole is not an integer, the lowest eigenvalue forthe Neumann and the Steklov problems is positive. We establish isoperimetricinequalitites for the lowest eigenvalue in the spirit of the classicalinequalities of Szeg\"o-Weinberger, Brock and Weinstock, the model domain beinga disk with the pole at its center. We consider more generally domains in theplane endowed with a rotationally invariant metric, which include the sphericaland the hyperbolic case.

Isoperimetric inequalities for the magnetic Neumann and Steklov problems with Aharonov-Bohm magnetic potential