Complete Hypersurfaces in a Real Space Form

Анотація

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 $c=1, k'\ge 2n\sqrt{n(n-1)}$, for any $i$, $\sum_{j\neq i} \lambda^2_j >n(n-1)$ and $|h|^2\le nH^2+(B^+_H)^2$, then $M^n$ is totally umbilical, or (i) $n\ge 3$, $M^n$ is locally an $H(r)$-torus with $r^2< \frac{n-1}{n}$, (ii) $n=2$, $M^n$ is locally an $H(r)$-torus with $r^2\neq \frac{n-1}{n}$; (2) if $c=0$ and $|h|^2\le nH^2+(\widetilde{B}^+_H)^2$, then $M^n$ is isometric to a standard round sphere, a hyperplane $R^n$ or $S^{n-1}(c_1)\times R^1$; (3) if $c=-1$ and $|h|^2\le nH^2+(\widehat{B}^+_H)^2$, then $M^n$ is totally umbilical or is isometric to $S^{n-1}(r)\times H^1(-1/(r^2+1))$ for some $r>0$, where $|h|^2$ denotes the squared norm of the second fundamental form of $M^n$, $B^+_H$, $\widetilde{B}^+_H$, and $\widehat{B}^+_H$, are denoted by (1.1), (1.2) and (1.3).

Mathematics Subject Classification: 53C40, 53C20.