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Math. Z. 227, 377–390 (1998) c Springer-Verlag 1998 1;? 2;?? Hongcang Yang , Qing-Ming Cheng Institute of Mathematics, Academia Sinica, 100080 Beijing, China Department of Mathematics, Josai University, Sakado, Saitama 350-02, Japan, e-mail: cheng@math.josai.ac.jp Received 21 April 1995; in final form 28 October 1996 1 Introduction The following conjecture is well known: Chern’s conjecture. For n -dimensional closed minimal hypersurfaces in the unit n +1 sphere S (1) with constant scalar curvature, the values S of the squared norm of the second fundamental forms should be discrete. For this conjecture, Simons [4], Chern et al [1] and Lawson [2] proved that the first and the second value are 0 and n , respectively. Yau [8] (or [9]) brought out this conjecture again as an open problem in his well-known problem section. On the third value of S , Peng and Terng [3] proved that if S > n , then (1-1) S > n + c(n ); where (1-2) c(n ) > 1=(12n ) is a positive constant. In addition, when n = 3, they got S 6. Hence they conjectured that the third value of S should be 2n since there exist E. Cartan’s n +1 isoparametric
Mathematische Zeitschrift – Springer Journals
Published: Mar 1, 1998
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