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

Nguyen Van Duc
대한수학회 (The Korean Mathematical Society)|2022.05|대한수학회보 (Bulletin of the KMS)|제59권 제3호|pp. 709-723|DOI 10.4134/BKMS.b210426

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키워드

Finite indexfiniteness resultharmonic formsminimal hypersurfacesrigidity theorem
주제분류수학

초록

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.

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.

참고문헌 (21)

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관련 논문AI추천

STABLE MINIMAL HYPERSURFACES WITH WEIGHTED POINCARE INEQUALITY IN A RIEMANNIAN MANIFOLDNguyen Dinh Sang, Nguyen Thi Thanh대한수학회 · 대한수학회논문집2014 · 제29권 제1호Structure of stable minimal hypersurfaces in a Riemannian manifold of nonnegative Ricci curvature김정진, 윤갑진대한수학회 · 대한수학회보2013 · 제50권 제4호On the structure of minimal submanifolds in a Riemannian manifold of non-negative curvature김동호, 윤갑진대한수학회 · 대한수학회보2009 · 제46권 제6호Symmetry and uniqueness of embedded minimal hypersurfaces in $\mathbb R^{n+1}$박성호대한수학회 · 대한수학회보2021 · 제58권 제1호Rigidity of minimal submanifolds with flat normal bundle서검교대한수학회 · 대한수학회논문집2008 · 제23권 제3호$L^2$ harmonic $1$-forms on submanifolds with weighted Poincar\'{e} inequalityXiaoli Chao, Yusha Lv대한수학회 · 대한수학회지2016 · 제53권 제3호

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