On the existence of minimal hypersurfaces with arbitrarily large area and Morse index (Q2108748)

A closed Riemannian manifold \((M^{n+1},g)\) is called \textit{bumpy} if there is no immersed closed minimal hypersurface with a non-trivial Jacobi field. By the work of \textit{B. White} [Am. J. Math. 139, No. 4, 1149--1155 (2017; Zbl 1379.53084)], the set of bumpy metrics is generic in the Baire sense. As the main result of the paper, the author showed that a bumpy closed Riemannian manifold \((M^{n+1},g)\) with \(3\leq n+1\leq 7\) admits a sequence of connected closed embedded two-sided minimal hypersurfaces whose areas and Morse indices both tend to infinity. This improves a result of \textit{O. Chodosh} and \textit{C. Mantoulidis} [Int. Math. Res. Not. 2021, No. 14, 10841--10847 (2021; Zbl 1484.53096)], where it was showed that there exists a sequence of connected closed embedded minimal hypersurfaces whose areas tend to infinity. The key of this result is the \textit{connected}. In fact, if one does not require connected, this follows from the following complete Morse-theoretic description of the set of min-max minimal hypersurfaces for a generic case, which was obtained by combining the work of \textit{O. Chodosh} and \textit{C. Mantoulidis} [Ann. Math. (2) 191, No. 1, 213--328 (2020; Zbl 1431.49045)], \textit{X. Zhou} [Ann. Math. (2) 192, No. 3, 767--820 (2020; Zbl 1461.53051)] and \textit{F. C. Marques} and \textit{A. Neves} [Adv. Math. 378, Article ID 107527, 59 p. (2021; Zbl 1465.53076)]: On a bumpy closed Riemannian manifold \((M^{n+1},g)\) with \(3\leq n+1\leq 7\). For each positive integer \(p\), there exists a smooth closed embedded multiplicity-one two-sided minimal hypersurface \(\Sigma_p\) such that \(\mathrm{index}(\Sigma_p)=p\) and the area satisfies \(\lim_{p\to\infty}\mathrm{area}(\Sigma_p)/p^{1/{(n+1)}}=c(n)\mathrm{vol}(M,g)^{n/{(n+1)}}\).