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The min-max theory for minimal surface is developed for the area functional on the space of cycle $Z_n(M)$, producing an unstable minimal surface with area equal to the width. Of course, this is powerful when we are in a manifold with trivial homology (most of the papers on the subject considered this case). On a closed $n+1$-dimensional manifold $M$ with non-trivial $H_n$, we can minimize the area inside a non-trivial homology class to obtain an area-minimizing hypersurface.

My question is, do people do min-max on manifold with non-trivial homology too? Like, if we only consider sweep-outs into a fixed homology class $[\alpha]\neq 0$: $\phi: X\rightarrow [\alpha] \subset Z_n(M)$ and then define the width in the similar manner, can we find an unstable minimal hypersurface in $[\alpha]$, providing that the width is greater than the area of the minimizing one in $[\alpha]$?

If this is known, is there a case where we can find the exact width of a homology class $\alpha$? Appreciate any lead or literature on this.

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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.

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