Existence of multiple closed CMC hypersurfaces with small mean curvature

Abstract

Let $(M^{n+1},g)$ be a closed Riemannian manifold, $n+1\geq 3$. We will prove that for all $m \in \mathbb{N}$, there exists $c^{*}(m)>0$, which depends on $g$, such that if $0<c<c^{*}(m)$, $(M,g)$ contains at least $m$ many closed $c$-CMC hypersurfaces with optimal regularity. More quantitatively, there exists a constant $γ_0$, depending on $g$, such that for all $c>0$, there exist at least $γ_0c^{-\frac{1}{n+1}}$ 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 $(M,g)$.

Theorems & Definitions (25)

• Theorem 1.1
• Theorem 1.2
• Definition 3.1
• Definition 3.2
• Theorem 3.3: bw1bw2bcw, zz1 when $3\leq n+1\leq 7$
• Definition 3.4
• Theorem 3.5: zz1zhou2
• proof
• Proposition 3.6
• Lemma 3.7