Eigenvalue estimate and gap theorems for submanifolds in the hyperbolic space

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• 作者：Hezi Lin ; ^{lhz1@fjnu.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：53C21 ; 53C42
• 刊名：Nonlinear Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：126-137
• 全文大小：636 K

文摘

Let Mⁿ$M^n$ be a complete non-compact submanifold in the hyperbolic space H^n+m${\mathbb{H}}^{n+m}$. We first give an estimate for the bottom of the spectral of the Laplace operator on Mⁿ$M^n$, under an integral pinching condition on the mean curvature. As a consequence of this estimation, we show some vanishing theorems for L²$L^2$ harmonic forms in certain degrees if the total mean curvature of Mⁿ$M^n$ is less than an explicit constant and its total curvature is less than a suitable related constant. In addition, we obtain some vanishing results under certain pointwise restrictions on the traceless second fundamental form. Moreover, according to the nonexistence of nontrivial L²$L^2$ harmonic 1$1$-forms, we can further prove some one-end theorems.