Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc.

Published online March 15, 2022

Copyright © The Korean Mathematical Society.

Finiteness and vanishing results on hypersurfaces with finite index in $\mathbb{R}^{n+1}$: a revision

Nguyen Van Duc

(VNU) University of Science, Hanoi, Vietnam National University, Hanoi.

Abstract

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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