摘要
等谱问题是微分几何中具有悠久历史同时也是十分重要的一个研究方向,该问题是:黎曼流形上的Laplace算子的谱是否决定它的几何?一般而言,这个问题的回答是否定的.因而该问题的研究就被分成两个部分:一是研究和刻画具有否定回答(等谱问题)的黎曼流形的例子和它们的性质;二是研究和刻画具有肯定回答(等谱问题)的黎曼流形的特征.本文中我们就单位球中的Willmore环面来研究它们的等谱性质.我们的结论如下:
设M是球面S~(n+1)(1)(n≥2)中闭Willmore超曲面,如果M满足下面两个条件:
(1)Spec~p(M)=Spec~p(W_(m,n-m)(P=0,1,2),其中W_(m,n-m)是S~(n+1)(1)中的Willmore环面;
(2)M与W_(m,n-m)有同样的常平均曲率.那么M=W_(m,n-m).
Isospectral problem is an important subject in geometry. The problem is: Does the geometry of a Riemannian manifold be decided by the spectrum of the Laplace operator of the manifold? In this paper we study the isospectral problem of Willmore hypersurfaces in (n+1)-dimensional sphere S~(n+1)(1). The conclusion is as follows:
Let M be a closed Willmore hypersurface in the sphere S(n+1)(1) (n≥2) with the same constant mean curvature of the Willmore torus W_(m,n-m)=S~(n-m)((?))×S~m((?)).If Spec~p(M)=Spec~p(W_(m,n-m)) (p= 0,1,2), then M=W_(m,n-m).
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