The structure of weakly stable CMC hypersurfaces with free boundary

Abstract: In this paper, we prove that a complete noncompact weakly stable free boundary CMC $H$-hypersurface $(M^{n},\partial M)$ properly immersed in $(N^ {n+1},\partial N)$ must have one end, provided that $N$ has bounded geometry and weakly convex boundary, satisfies $\inf\Ric_{N}>-\frac{1}{n}H^2$ and $\biRic_{N}\geq \frac{(n-5)}{4}H^2$. Secondly, we establish a non-existence result for noncompact free boundary CMC hypersurfaces under certain conditions, as detailed in Theorem \ref{thm 4}. Finally, we give a rigidity theorem for free boundary minimal hypersurfaces in $5$-manifolds.