Existence of minimal hypersurfaces with arbitrarily large area and possible obstructions (Q6560733)

Yau's conjecture (proved by \textit{A. Song} [Ann. Sci. Éc. Norm. Supér. (4) 56, No. 4, 1085--1134 (2023; Zbl 1542.53018)] yields the existence of infinitely many minimal hypersurfaces in a \((n+1)\)-Riemannian manifold \((M,g)\) with \(3\leq n+1\leq 7\). Under these assumptions on \((M,g)\), the authors show that either\N\begin{enumerate}\N\item[(1)] \((M,g)\) has minimal hypersurfaces of arbitrarily large area\N\item[(2)] there exist in \((M,g)\) uncountably many connected stable minimal hypersurfaces of uniformly bounded area and with infinitely many distinct areas.\N\end{enumerate}\NThey construct an example for case 2; nevertheless case 2 has a Cantor set structure which is a very pathological and cannot exist in some manifolds. For example, if \((M,g)\) is real analytic, it always verifies 1.\N\NThe proof uses (quoting the abstract): the Almgren-Pitts min-max theory proposed by Marques-Neves, see for example [\textit{F. C. Marques} and \textit{A. Neves}, Adv. Math. 378, Article ID 107527, 59 p. (2021; Zbl 1465.53076)], the ideas developed by Song in his proof of Yau's conjecture, and the resolution of the generic multiplicity-one conjecture by \textit{X. Zhou} [Ann. Math. (2) 192, No. 3, 767--820 (2020; Zbl 1461.53051)].\N\NUsing [\textit{O. Chodosh} et al., Invent. Math. 209, No. 3, 617--664 (2017; Zbl 1378.53072)], the authors derive the existence of connected minimal surfaces of arbitrarily large area and index in closed analytic Riemannian \(3\)-manifolds with positive scalar curvature.