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《Journal of Yunnan Normal University(Natural Sciences Edition)》 2008-01
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A Theorem on the First Eigencalue of the Laplacian for Compact Submanifolds of Euclidean Space

YANG Hui-Zhang1,Guo Shun-Zi2(1.Department of Mathematics,Hong He Teacher's College,Yunnan Mengzi 661100,China;2.Department of Mathenatice,Zhang Zhou Teacher's College,Zhangzhou 363000,China)  
In this paper,we give a global theorem:let x:Mn→Rn+p is immersion of a compact manifold M into Euclidean space R.let F(q) is the maxinum of the square of the norm of the second fundamental form along unit normal vector u,λ1 is the first eigencalue of Laplacian operator △,then we have equality if and only if x immerses M as a hypersphere in an(n+1) —dimensional linear sub-space of Rn+p.So we generalize R.C.Reillys result in[1].
【Fund】: 红河学院硕士博士校外项目课题(ZSS06006)
【CateGory Index】: O186
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Chinese Journal Full-text Database 8 Hits
1 Shin-Jye Liang Jyh-Haw Tang Ming-Shun Wu Department of Marine Environmental Informatics,National Taiwan Ocean University,Keelung 20224,Taiwan,China Department of Bioenvironmental Engineering,Chung-Yuan Christian University,Chung-Li 32023,Taiwan,ChinaDepartment of Geographic Information Systems,Feng Chia University,Taichung 40724,Taiwan,China;Solution of shallow-water equations using least-squares finite-element method[J];力学学报(英文版);2008-05
2 Ru Shan WANG College of Mathematics and Computer Science,Anhui Normal University,Wuhu 241000,P.R.China and School of Mathematical Sciences,Fudan University,Shanghai 200433,P.R.China;Sharp Upper Bounds for λ_1~(L_r) of Immersed Hypersurfaces and Their Stability in Space Forms[J];数学学报(英文版);2008-05
3 GE JianQuan Department of Mathematical Sciences,Tsinghua University,Beijing 100084,China;On mean curvatures in submanifolds geometry[J];中国科学(A辑:数学)(英文版);2008-06
4 Wang Rushan (Anhui Normal University,Wuhu,241000);Upper Bound Estimates of λ_(1~r)~L on Closed Hypersurfaces in E~(n+1)[J];Mathematical Theory and Applications;2008-03
5 GUO Shun-zi(School of Mathematics,Yunnan Normal University,Kunming 650092,China);A Theorem for Hige Order Ccnformal Mean Curvature on Supresurface and Its Application[J];Journal of Honghe University;2006-02
6 XU Ling & GE JianQuan School of Mathematical Sciences,Laboratory of Mathematics and Complex Systems,Beijing Normal University,Beijing 100875,China;Variational formulas of higher order mean curvatures[J];中国科学:数学(英文版);2012-10
7 GUO Shun-zi1,HE Guo-xiang2(1.Department of Mathematics,Yunnan Normal University,Kunming 650092,China;2.Pantiange High School,Weixi 674602,China);A Theorem for High Order Conformal Mean Curvature on Supersurface[J];Journal of Sichuan University of Science & Engineering(Natural Science Edition);2006-04
8 WANG Rushan~* *School of Mathematical Sciences,Fudan University,Shanghai 200433,China;Department of Mathematics,Anhui Normal University,Wuhu 241000,Anhui,China.;First Eigenvalue of Compact Submanifolds Without Boundary in Space Forms[J];Chinese Annals of Mathematics;2007-04
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