A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space
A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space
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DOI：
10.4171/cmh/88
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发表时间：
2006-09
影响因子：
0.9
通讯作者：
B. Colbois;Jean Grosjean
中科院分区：
文献类型：
--
作者：
B. Colbois;Jean Grosjean
本文给出了欧氏空间紧超曲面上拉普拉斯算子的第一个非零本征值的Pinch定理。的确，我们证明了如果$M$的体积是$1$，则对于任何$varepsilon>；0$，存在一个常数$C_{\varepsilon}$依赖于$M$的维度$n$和平均曲率$H$的$L$范数，使得如果$H$的$L_{2p}$-范数$H2p}$($p2$)满足$n\|H|{2p}-C{varepsilon}<；则$M$与半径为$(n/\lambda)^{1/2}$的圆球之间的Hausdorff距离小于$\varepsilon$。进一步地，我们证明了如果$C$是一个依赖于$n$和第二基本形式的$L范数的足够小的常数，则Pinching条件$n H2p-C<；\la$暗示$M$微分同胚于$n$维球面.
In this paper, we give pinching Theorems for the first nonzero eigenvalue $\lambda$ of the Laplacian on the compact hypersurfaces of the Euclidean space. Indeed, we prove that if the volume of $M$ is $1$ then, for any $\varepsilon>0$, there exists a constant $C_{\varepsilon}$ depending on the dimension $n$ of $M$ and the $L_{\infty}$-norm of the mean curvature $H$, so that if the $L_{2p}$-norm $\|H\|_{2p}$ ($p\geq 2$) of $H$ satisfies $n\|H\|_{2p}-C_{\varepsilon}<\lambda$, then the Hausdorff-distance between $M$ and a round sphere of radius $(n/\lambda)^{1/2}$ is smaller than $\varepsilon$. Furthermore, we prove that if $C$ is a small enough constant depending on $n$ and the $L_{\infty}$-norm of the second fundamental form, then the pinching condition $n\|H\|_{2p}-C<\la$ implies that $M$ is diffeomorphic to an $n$-dimensional sphere.