Existence of multiple closed CMC hypersurfaces with small mean curvature

Abstract

Min-max theory for constant mean curvature (CMC) hypersurfaces has been developed by Zhou–Zhu $[\href{https://doi.org/10.1007/s00222-019-00886-1}{39}]$ and Zhou $[\href{https://doi.org/10.4007/annals.2020.192.3.3}{38}]$. In particular, in $[\href{https://doi.org/10.1007/s00222-019-00886-1}{39}]$, Zhou and Zhu proved that for any $c \gt 0$, every closed Riemannian manifold $(M^{n+1}, g), 3 \leq n + 1 \leq 7$, contains a closed $c$-CMC hypersurface. In this article we will show that the min-max theory for CMC hypersurfaces in $[\href{https://doi.org/10.1007/s00222-019-00886-1}{39}, \href{https://doi.org/10.4007/annals.2020.192.3.3}{38}]$ can be extended in higher dimensions using the regularity theory of stable CMC hypersurfaces, developed by Bellettini–Wickramasekera $[\href{https://doi.org/10.48550/arXiv.1802.00377}{4}, \href{https://doi.org/10.48550/arXiv.1902.09669}{5}]$ and Bellettini–Chodosh–Wickramasekera $[\href{https://doi.org/10.1016/j.aim.2019.05.023}{3}]$. Furthermore, we will prove that the number of closed $c$-CMC hypersurfaces in a closed Riemannian manifold $(M^{n+1}, g), n+1 \geq 3$, tends to infinity as $c \to 0^+$. More quantitatively, there exists a constant $\gamma_0$, depending on $g$, such that for all $c \gt 0$, there exist at least $\gamma_0 c^{-\frac{1}{n+1}}$ many closed $c$-CMC hypersurfaces (with optimal regularity) in $(M,g)$.