Spherical rigidities of submanifolds in Euclidean spaces

Abstract

In this paper, we study $n$-dimensional complete immersed submanifolds in a Euclidean space $E^{n+p}$. We prove that if $M^n$ is an $n$-dimensional compact connected immersed submanifold with nonzero mean curvature $H$ in $E^{n+p}$ and satisfies either:

(1)$S\le \frac{n^2{H}^2}{n-1},$ or

(2)$n^2{H}^2\le \frac{(n-1)R}{n-2}$,

then $M^n$ is diffeomorphic to a standard $n$-sphere, where $S$ and $R$ denote the squared norm of the second fundamental form of $M^n$ and the scalar curvature of $M^n$, respectively.

On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere $S^n\left(c\right)$, the totally geodesic Euclidean space $E^n$, and the generalized cylinder $S^{n-1}\left(c\right)\times E^1$ are only $n$-dimensional $(n>2)$ complete connected submanifolds $M^n$ with constant mean curvature $H$ in $E^{n+p}$ if $S\le n^2{H}^2/(n-1)$ holds.