Existence of minimal hypersurfaces with arbitrarily large area and possible obstructions

TL;DR

This paper establishes a dichotomy in closed Riemannian manifolds of dimension 3 to 7, showing either the existence of minimal hypersurfaces with arbitrarily large area or uncountably many stable ones, with applications to analytic manifolds.

Contribution

It proves a new dichotomy for minimal hypersurfaces in certain manifolds, linking large area hypersurfaces to the abundance of stable ones, using advanced min-max and genericity theories.

Findings

Either large area minimal hypersurfaces exist or uncountably many stable ones do.

The stable hypersurfaces can have a complex Cantor set structure.

Existence of large area minimal hypersurfaces in analytic manifolds.

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.