Existence of minimal hypersurfaces with arbitrarily large area and possible obstructions

Abstract

We prove that in a closed Riemannian manifold with dimension between $3$ and $7$, either there are minimal hypersurfaces with arbitrarily large area, or there exist uncountably many stable minimal hypersurfaces. Moreover, the latter case has a very pathological Cantor set structure which does not show up in certain manifolds. Among the applications, we prove that there exist minimal hypersurfaces with arbitrarily large area in analytic manifolds. In the proof, we use 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.

31 pages, 3 figures. The title is revised to describe the results more accurately. Final version. To appear in the Journal of Functional Analysis