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

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.

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}})

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 $\frac{(n-1)^2(1-|H|^2)}{4}$.