Spherical rigidities of submanifolds in Euclidean spaces

Dedicated to Professor Buchin Su for his 100th birthday

抄録

In this paper, we study n-dimensional complete immersed submanifolds in a Euclidean space \bm{E}^n+p. We prove that if Mⁿ is an n-dimensional compact connected immersed submanifold with nonzero mean curvature H in \bm{E}^n+p and satisfies either:
(1) s\displaystyle ≤\frac{n²H²}{n-1}, or
(2) n²H²\displaystyle ≤\frac{(n-1)R}{n-2}, then Mⁿ is diffeomorphic to a standard n-sphere, where S and R denote the squared norm of the second fundamental form of Mⁿ and the scalar curvature of Mⁿ 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 Sⁿ(c), the totally geodesic Euclidean space \bm{E}ⁿ, and the generalized cylinder S^n-1(c)× \bm{E}¹ are only n-dimensional (n>2) complete connected submanifolds Mⁿ with constant mean curvature H in \bm{E}^n+p if S≤ n²H²/(n-1) holds.