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 [7]. 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

Supported by: This research is funded by the University of Science, Vietnam National University, Hanoi under project number TN.21.04. The author was partially supported by Vingroup Innovation Foundation VINIF under grant number VINIF.2019. ThS.18.