EMBEDDED HYPERSURFACES WITH CONSTANT mTH MEAN CURVATURE IN A UNIT SPHERE

GUOXIN WEI, QING-MING CHENG, HAIZHONG LI
2010 Communications in Contemporary Mathematics  
In this paper, we study n-dimensional hypersurfaces with constant mth mean curvature in a unit sphere S n+1 (1) and construct many compact nontrivial embedded hypersurfaces with constant mth mean curvature Hm > 0 in S n+1 (1), for 1 ≤ m ≤ n − 1. Moreover, if the 2nd mean curvature H 2 takes value between 1 (tan π k ) 2 and k 2 −2 n for any integer k ≥ 2 and n ≥ 3, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H 2 (i.e. constant scalar curvature) in S
more » ... +1 (1); If the 4th mean curvature H 4 takes value between 1 (tan π k ) 4 and k 4 −4 n(n−4) for any integer k ≥ 3 and n ≥ 5, then there exists an n-dimensional compact nontrivial embedded hypersurface with constant H 4 in S n+1 (1). S l (a)× S n−l (b), 1 ≤ l ≤ n− 1 are compact embedded hypersurfaces in a unit sphere S n+1 (1). Hence, it is natural to ask the following: Question. Do there exist compact embedded hypersurfaces with constant mth mean curvature H m in S n+1 (1) other than the standard round spheres and Clifford hypersurfaces? When m = 1, namely, when the mean curvature is constant, Ripoll ([13]) has proved the existence of compact embedded hypersurfaces of S 3 (1) with constant mean curvature (H = 0, ± √ 3 3 ) other than the standard round spheres and the Clifford hypersurfaces. Then, Brito-Leite ([2]) have proved that there exist compact embedded hypersurfaces with constant mean curvature H in S n+1 (1), which are not isometric to the standard round spheres and the Clifford hypersurfaces. Recently, Perdomo ([12]) has proved that there exists an n-dimensional compact nontrivial embedded hypersurface with constant mean curvature H > 0 in S n+1 (1) if mean curvature H takes value between
doi:10.1142/s0219199710004081 fatcat:vk3e7q5ro5ethaytosrzmyxm6y