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