Existence of multiple closed CMC hypersurfaces with small mean curvature

Abstract

Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\ge 3$. We will prove that for all $m\in \mathbb{N}$, there exists $c^{\ast }(m)>0$, which depends on $g$, such that if $0<c<c^{\ast }(m)$, $(M,g)$ contains at least $m$ many closed $c$-CMC hypersurfaces with optimal regularity. More quantitatively, there exists a constant ${\gamma }_0$, depending on $g$, such that for all $c>0$, there exist at least ${\gamma }_0{c}^{-\frac{1}{n+1}}$ many closed $c$-CMC hypersurfaces (with optimal regularity) in $(M,g)$. This extends the theorem of Zhou and Zhu, where they proved the existence of at least one closed $c$-CMC hypersurface in $(M,g)$.

v2: Improved exposition, added references

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