L2 harmonic 2-forms on a hypersurface in Euclidean space
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摘要: 研究欧氏空间${\bf{R}}^{n+1}(n\geqslant 3)$中完备超曲面$M$上的$L^2$调和$2$-形式.应用Bochner技巧, 证明了当$M$的无迹对称张量$\Phi$和平均曲率向量$H$的$L^n(M)$范数均有只依赖于$n$的适当上界时, $M$上的$L^2$调和$2$-形式是平行的.进一步, 若$M$为非极小超曲面, 则$M$上不存在非平凡的$L^2$调和$2$-形式.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 $M$.
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Key words:
- Euclidean space /
- hypersurface /
- L2 harmonic 2-forms /
- non-minimal
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