Finding Large Minimal Hypersurfaces

We show that for any closed Riemannian manifold with dimension between 3 and 7, either there are minimal hypersurfaces with arbitrarily large area, or the space of certain pathological-looking minimal hypersurfaces has a Cantor set structure. In particular, among the applications, we prove that there exist minimal hypersurfaces with arbitrarily large area in analytic manifolds. The proof uses the Almgren-Pitts min-max theory proposed by Marques-Neves, the ideas developed by Song in his proof of Yau's conjecture, and the resolution of the generic multiplicity-one conjecture by Zhou.