We will discuss the following problem: given a closed $3$-manifold $M$ and $H \geq 0$, is there a surface with constant mean curvature $H$ in $M$ whose genus $g$ is controlled? We will show that for almost every $H$, one can construct a branched immersion with these properties, with $g$ bounded above by the Heegaard genus of $M$. The proof relies on a min-max construction based on a perturbation of the area functional involving the second fundamental form of the immersion, introduced by T. Rivière. This is joint work with Xuanyu Li (Cornell).