Complete Hypersurfaces in a Real Space Form - Shichang, Shu (Журнал математичної фізики, аналізу, геометрії, 2008)
Let M^n be an n-dimensional complete hypersurface with the scalar curvature n(n-1)R and the mean curvature H being linearly related, that is, n(n-1)R=k’H (k’>0) in a real space form R^n+1(c). Assume that the mean curvature is positive and obtains its maximum on M^n. We show that (1) if c=1, k’2nn(n-1), for any i, _j eq i λ^2_j >n(n-1) and |h|^2nH^2+(B^+_H)^2, then M^n is totally umbilical, or (i) n3, M^n is locally an H(r)-torus with r^2< n-1n, (ii) n=2, M^n is locally an H(r)-torus with r^2 eq n-1n; (2) if c=0 and |h|^2nH^2+(B^+_H)^2, then M^n is isometric to a standard round sphere, a hyperplane R^n or S^n-1(c_1)× R^1; (3) if c=-1 and |h|^2nH^2+(B^+_H)^2, then M^n is totally umbilical or is isometric to S^n-1(r)× H^1(-1/(r^2+1)) for some r>0, where |h|^2 denotes the squared norm of the second fundamental form of M^n, B^+_H, B^+_H, and B^+_H, are denoted by (1.1), (1.2) and (1.3). Mathematics Subject Classification: 53C40, 53C20.