the vector bundle on which $\Delta_A$ acts. In both cases, implicitly, the flux of the potential $A$ plays a crucial role.

[$\bullet\quad$]{}From now on we will denote by $\lambda_1(\Omega,A)$ the first eigenvalue of $\Delta_A$ on $(\Omega,g)$.

Main lower bound
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Our lower bound is partly inspired by the results in [@HHHO] for plane domains. First, recall that if $c$ is a closed parametrized curve (a loop), the quantity: $$\Phi^A_c=\dfrac{1}{2\pi}\oint_{c}A$$ is called the [*flux*]{} of $A$ across $c$. (We assume that $c$ is travelled once, and we will not specify the orientation of the loop, so that the flux will only be defined up to sign: this will not affect any of the statements, definitions or results which we will prove in this paper). Let then $\Omega$ be a fixed plane domain with one hole, and let $\Phi^A$ be the flux of the harmonic potential $A$ across the inner boundary curve. In Theorem 1.1 of [@HHHO] it is first remarked that $\lambda_1(\Omega, A)$ is positive if and only if $\Phi^A$ is not an integer (but see the precise statement in Section 2.1 below). Then, it is shown that $\lambda_1(\Omega,A)$ is maximal precisely when $\Phi^A$ is congruent to $\frac 12$ modulo integers. 