Bull. Korean Math. Soc. 2022; 59(3): 709-723
Online first article May 31, 2022 Printed May 31, 2022
https://doi.org/10.4134/BKMS.b210426
Copyright © The Korean Mathematical Society.
Nguyen Van Duc
Vietnam National University
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.
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