Bull. Korean Math. Soc.
Published online March 15, 2022
Copyright © The Korean Mathematical Society.
Nguyen Van Duc
(VNU) University of Science, Hanoi, Vietnam National University, Hanoi.
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.
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