There is a special case of this question that has been studied. If $\Sigma$ is a surface and $\psi:\Sigma\to \Sigma$ a pseudo-Anosov mapping class, then it is known that the mapping torus $T_\psi$ admits a unique hyperbolic metric. In this case, the fiber $\psi$ is isotopic to a minimal surface. One can also perform a min-max for the surface moving around the fibration over the circle, and ask whether this is strictly larger that the minimum? If it is equal, then the manifold is foliated by minimal surfaces. It is believed that this is not possible (I think this was conjectured by Uhlenbeck), and there are partial results to this effect.

There is a special case of this question that has been studied. If $\Sigma$ is a surface and $\psi:\Sigma\to \Sigma$ a pseudo-Anosov mapping class, then it is known that the mapping torus $T_\psi$ admits a unique hyperbolic metric. In this case, the fiber $\psi$ is isotopic to a minimal surface. One can also perform a min-max for the surface moving around the fibration over the circle, and produce at least two minimal surfaces isotopic to the fiber (Sacks-Uhlenbeck, Corollary 5.5). One may ask whether the maximum is strictly larger that the minimum? If it is equal, then the manifold is foliated by minimal surfaces. It is believed that this is not possible (I think this was conjectured by Uhlenbeck), and there are partial results to this effect.