Let $(\Sigma,g)$ be a smooth genus $k$ closed oriented surface endowed with a Riemannian metric $g$. Let $P_\Sigma$ be a (necessarily) trivial $\mathrm{SU}(2)$-principal bundle over $\Sigma$. Let $\mathcal{A}_\Sigma$ be the space of connexions on $P_\Sigma$. Define $S:\mathcal{A}_\Sigma\to \mathbb R_{\geq 0}$ be the Yang-Mills functional $S(A):=\frac12\|F_A\|^2_{L^2}$ where $F_A\in \Omega^2(\Sigma;\mathrm{AdP}_\Sigma)$ is $A$'s curvature 2-form. It is well known that the critical set of $S$ is given by : $$ \mathrm{crit}(S) = \{A\in \mathcal{A}_\Sigma | \delta_A F_A = 0\} $$ where $\delta_A$ is the covariant codifferential (a.k.a $\mathrm{d}_A^*$ the $L^2$-adjoint of the covariant exterior derivative $\mathrm{d}_A$).
Question : is there an $m\in \mathbb R_{>0}$ such that $$ \mathrm{crit}(S) \cap \{A\in \mathcal{A}_\Sigma | 0<S(A)<m\} = \emptyset \qquad \text{?} $$