| 付海平,陶永芊.双曲空间中完备子流形的特征值估计[J].数学研究及应用,2013,33(5):598~606 |
| 双曲空间中完备子流形的特征值估计 |
| Eigenvalue Estimates for Complete Submanifolds in the Hyperbolic Spaces |
| 投稿时间：2012-05-29 修订日期：2013-02-19 |
| DOI：10.3770/j.issn:2095-2651.2013.05.009 |
| 中文关键词: 有限$L^q$范数曲率 第一特征值 双曲空间 稳定超曲面. |
| 英文关键词:finite $L^q$ norm curvature first eigenvalue hyperbolic space stable hypersurface. |
| 基金项目:国家自然科学基金(Grant No.11261038),江西省自然科学基金(Grant Nos.2010GZS0149;20132BAB201005), 江西省教育厅青年基金(Grant No.GJJ11044). |
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| 中文摘要: |
| 研究了$(n+p)$维双曲空间$\mathbb{H}^{n+p}$中完备非紧子流形的第一特征值的上界.特别地,证明了$\mathbb{H}^{n+p}$中具有平行平均曲率向量$H$和无迹第二基本形式有限$L^q(q\geq n)$范数的完备子流形的第一特征值不超过$\frac{(n-1)^2(1-|H|^2)}{4}$,和$\mathbb{H}^{n+1}(n\leq5)$中具有常平均曲率向量$H$和无迹第二基本形式有限$L^q(2(1-\sqrt{\frac{2}{n}})
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| 英文摘要: |
| In this paper, we study upper bounds of the first eigenvalue of a complete noncompact submanifold in an $(n+p)$-dimensional hyperbolic space $\mathbb{H}^{n+p}$. In particular, we prove that the first eigenvalue of a complete submanifold in $\mathbb{H}^{n+p}$ with parallel mean curvature vector $H$ and finite $L^q(q\geq n)$ norm of traceless second fundamental form is not more than $\frac{(n-1)^2(1-|H|^2)}{4}$. We also prove that the first eigenvalue of a complete hypersurfaces which has finite index in $\mathbb{H}^{n+1}(n\leq 5)$ with constant mean curvature vector $H$ and finite $L^q(2(1-\sqrt{\frac{2}{n}})< q<2(1+\sqrt{\frac{2}{n}}))$ norm of traceless second fundamental form is not more than $\frac{(n-1)^2(1-|H|^2)}{4}$. |
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