Bulletin of the
Korean Mathematical Society BKMS

In this note, we revise some vanishing and finiteness results on hypersurfaces with finite index in $\mathbb{R}^{n+1}$. When the hypersurface is stable minimal, we show that there is no nontrivial $L^{2p}$ harmonic $1$-form for some $p$. The our range of $p$ is better than those in \cite{DS}. With the same range of $p$, we also give finiteness results on minimal hypersurfaces with finite index.

Keywords: Finite index; Finiteness result; Harmonic forms; Minimal hypersurfaces; Rigidity theorem.

MSC numbers: Primary 53C24; Secondary 53C40, 53A05.