<i>L</i><sup>2</sup> harmonic 2-forms on a hypersurface in Euclidean space

ZHANG Quan-rui; LIU Jian-cheng

doi:10.3969/j.issn.1000-5641.2018.03.005

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ZHANG Quan-rui, LIU Jian-cheng. L2 harmonic 2-forms on a hypersurface in Euclidean space[J]. Journal of East China Normal University (Natural Sciences), 2018, (3): 38-45. doi: 10.3969/j.issn.1000-5641.2018.03.005

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ZHANG Quan-rui, LIU Jian-cheng. L² harmonic 2-forms on a hypersurface in Euclidean space[J]. Journal of East China Normal University (Natural Sciences), 2018, (3): 38-45. doi: 10.3969/j.issn.1000-5641.2018.03.005

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L² harmonic 2-forms on a hypersurface in Euclidean space

doi: 10.3969/j.issn.1000-5641.2018.03.005

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Received Date: 2017-05-01
Publish Date: 2018-05-25

Abstract

Abstract

In this paper, we study $L^2$ harmonic $2$-forms on a complete hypersurface $M$ of Euclidean space ${\bf{R}}^{n+1} (n\geqslant3)$. By applying the Bochner technique, we prove that if the $L^n(M)$ norms of the traceless second fundamental form $\Phi$ and the mean curvature vector $H$ are both bounded from above by certain constants which depend only on $n$, then the $L^2$ harmonic $2$-forms on $M$ are parallel. Furthermore, if $M$ is a non-minimal hypersurface, then there is no nontrivial $L^2$ harmonic $2$-form on $M$.
- Euclidean space,
- hypersurface,
- L² harmonic 2-forms,
- non-minimal

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References(12)

References

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